ISSN:0301-5696    

DOI:10.1039/FS98419FX001

出版商:RSC    

年代:1984

数据来源: RSC

摘要:

Faraday Symp.Chem. Soc. 1984 19 7-15 General Introduction Computational Quantum Chemistry -1984 BY ERNESTR. DAVIDSON Indiana University Bloomington Indiana 47405 U.S.A. Received 12th December 1984 Over the course of the last three decades theoretical chemists have devised a quantitative model of the chemical bond which in spite of its increasing applicability is still often disregarded by experimentalists. Application of this model differs from the pre-quantum tradition of interp- olating between established experimental facts using human reasoning alone in that it relies on digital computers to extract the implications of the model for a particular molecule. While the improved accuracy associated with this model has resulted in increasingly complicated highly correlated wavefunctions a qualitative understanding of the model’s predictions can usually be obtained from a consideration of the simpler zeroth-order wavefunction.G. H. Hardy,l in an essay entitled ‘Apologies of a Pure Mathematician’ explained his motivation for wasting his life in such a useless activity as pure mathematics. Basically he sought in the theorems he created the same kinds of beauty and permanence which are usually attributed to great works of art or music. Many chemists also regard computational quantum chemistry as essentially useless. Worse the results obtained by computation are somehow regarded as uglier and less creative than those obtained by pure reasoning.** Certainly computational results are less permanent as improved calculations on simple molecules seem to appear monthly.In reply to these criticisms I would claim that through computation quantum chemists have created a quantitative model of the chemical bond which is beautiful to those who understand it and which is likely to be permanent. Further this develop- ment would never have been possible by reasoning alone. In this introductory lecture I want to address what I feel are the essential differences between this model and the traditional chemical models of bonding. I also wish to illustrate the range of validity of the model by giving examples of its successes and failures. The present quantitative model of the chemical bond is the result of 30 years of hard work by numerous chemists and applied mathematicians.In spite of its sweeping importance to chemistry the model itself is little appreciated by many chemists who continue to interpret their experimental results in terms of quite crude older models. Partly this is because the literature produced by computational chemists is so specialized and technical that most chemists are unaware that a true revolution has taken place. However it is also true that this new model requires use of computers to extract its implications for any given molecule so experimental chemists are less likely to adopt this model when analysing their data. From the beginnings of the atomic theory of matter chemists have been concerned with what held atoms together in molecules. The pre-quantum theory culminated in the work of G. N. Lewis and others which recognized concepts such as the shared electron pair the octet rule electronegativity partially ionic bonds etc.These models 7 GENERAL INTRODUCTION and the known periodic trends of the elements allowed qualitative predictions and interpretations of experimental data based on pure human reasoning. After the discovery of the Schrodinger equation the discovery of the electron spin and the formulation of the Pauli principle in terms of Slater determinants these pre-quantum ideas were transcribed into statements about wavefunctions. This approach was set forth most elegantly by Pauling in his book The Nature of the Chemical Bond,4 which became the Bible of a generation of inorganic chemists. This work continued the grand tradition of making qualitative predictions about chemical behaviour by using only human reasoning to interpolate between established experi- mental facts.It also made semiquantitative predictions of heats of reaction bond lengths etc. using simple formulae with a vague quantum-mechanical justification to interpolate between known quantitative data from other molecules. During this era of 1930-1950 two other models of the chemical bond were introduced. Crystal-field theory5 was developed to explain the perturbation of atomic energy levels by a crystal environment. This model led naturally to the ligand-field model of the perturbation of the d-electron energies in a transition-metal complex. Unlike the models discussed above this model required at least a calculator to solve the resultant eigenvalue problem.The resultant Tanabe-Sugano diagrams are still widely used to interpret spectra and ligand-metal bond energies. Another spin-off from the physicists’ approach to crystal structure was the molecular-orbital model of the chemical bond advocated by Mullikeq6 Huckel,’ WalshS and others. As a qualitative tool this model also allowed prediction of bond length and energy trends. It was more successful than the valence-bond method in describing molecular spectra but probably less successful in qualitatively accounting for the shape of molecules. With the advent of computers attempts were made to extend the applicability of the molecular-orbital model and to improve its reliability. The valence-boqd model was more difficult to reduce to a simpler computer algorithm so it was largely abandoned in quantitative applications.The present extended Huckel theory used by H~ffmann,~ CNDO developed by PoplelO and MNDO developed by Dewarl’ are the results of this refinement of the empirical molecular-orbital model. Embedded in these computer programs is a model of the chemical bond and a set of parameters derived by comparing results with experimental data. The model however is now complex enough that semiquantitative predictions require the use of a computer to work out the predictions of this model for any particular molecule. From the conception of the Schrodinger equation,12 there has been another group of chemists and physicists who were interested in really solving this equation.Heitler and London13 quickly carried out calculations for the hydrogen molecule. Hartree14 developed numerical SCF methods for atoms. Hylleraas15 and James and Coolidge16 carried out very-large-scale calculations (as measured by the number of hours of computing machine time used) on the helium atom and the hydrogen molecule. These calculations served mostly to establish that the Schrodinger equation unlike the Bohr theory which preceded it was probably ‘correct’ to at least one part in ten thousand for these two-electron systems. Application to larger systems was cut short by lack of adequate computational tools and the intervention of the Depression and World War 11. With access to computers in the 1950s a few chemists returned to this effort to accurately solve the Schrodinger equation and began developing computer programs to carry out ab initio calculations.As an aside let me pause at this point to say that I want to use ab initio in a very narrow sense in this lecture. For the purpose of this talk ab initio will mean ‘from the beginning’ i.e. without any input from past E. R. DAVIDSON Table 1. Results using atomic orbitalsav reaction AE,,,,/kJ mol-l AE,,,,/kJ mol-l H +2H 280 456 HF+H+F 209 569 CH +C+4H 1105 1636 CH +CH,+H 364 43 1 a Full CI experimental geometry minimum basis of near Hartree-Fock atomic orbitals. For H an SCF calculation in this basis gives AE = 209 kJ mol-l. MCSCF gives an extra 71 kJ mol-1 and distortion of the atomic orbitals gives 117 kJ mol-l.Correlation effects not describable with two distorted atomic orbitals give the remaining 59 kJ mol-l. experience or any parameters chosen because they worked for other atoms or molecules. With this definition true ab initio calculations are always exploratory and of interest mainly to other quantum chemists. If successful they may lead to improvements which can be incorporated into widely used ‘calibrated ab initio’ calculations carried out by applied quantum chemists. Like semiempirical methods these calculations assume transferability of parameters and methods between molecules and do not check the convergence of every result with respect to quality of wavefunction. The first ab initio calculations merely evaluated exactly the energies associated with the model wavefunctions of the semiempirical models.These calculations were disasters and led to a major schism between pure and empirical theorists2 which still has not healed. To oversimplify the situation these calculations gave energies which were in serious disagreement with experiment. The empiricists regarded this as proof that the ab initio approach was a waste of time. The purists on the other hand felt that their calculations proved that the empirical models were without fundamental justification and hence were basically ‘wrong’ even if they did produce useful predictions. Table 1 shows this dichotomy. While the binding in simple molecules like H, HF and CH is usually ‘explained’ using atomic orbitals actual ab initio calculations with Hartree-Fock atomic orbitals only account for 40-60% of the bond energy.It is recognized today of course that models like MNDO which seem to use atomic orbitals actually are parametrized to account for the effect of a more extended basis set. During the years from 1961 to 1970 a few theorists with access to computers persisted in their efforts to develop an accurate method based firmly on the Schrodinger equation for making predictions of chemical facts. The hoped-for method was required to give not only ‘right’ answers for ‘right’ reasons but was also required to be computationally tractable conceptually simple and widely applicable. By and large the poor results produced by many of the suggested methods led to this whole enterprise and the chemists associated with it being held in ill repute by the rest of the chemical community.Accurate results were slow in coming and when available gave little chemical insight. The requirement of tractibility was a major bottleneck in early calculations. As the power of computers has increased and the costs have gone down this requirement has eased considerably. Anyone carrying out today an MCSF gradient optimization of a transition-state geometry for a reaction like the Cope rearrangement” must be aware that most of the ideas required for this calculation were in place by 1969 but GENERAL INTRODUCTION Table 2. Approximate characteristics of some typical computers computer speed memory costa cost/speed IBM 650 1 103 1O6 1O6 CDC 1604 102 104 107 105 IBM 7094 104 104 107 103 CDC 7600 106 105 107 10 IBM 370 1O6 1O6 107 10 CRAY 1 107 1O6 107 1 VAX 11/780 105 1O6 105 1 FPS-164 106 1O6 105 lo-’ CRAY-XMP 1O8 107 107 10-1 a Cost new in 1984 U.S.dollars. such a calculation was unthinkable at that time because of the computer limitations. Table 2 summarizes this trend in computer costs. The major price break associated with the super-mini computers and the class VI computers has spurred the very ambitious calculations of the last decade. Wide applicability is also a limiting consideration. It is relatively easy to construct good basis sets for atoms and diatomic molecules which cannot be used for polyatomics. Use of elliptical orbitals for example gave excellent resultsl89 l9 for H and LiH but were not even applicable to heavier diatomics.Explicit use of the interelectron distance in the wavefunction also gave excellent resultsl5? l6 for H and He but is not widely useful for polyatomic molecules. Wide applicability requires an open-ended method which is capable at least in principle of arbitrary accuracy for any molecule. Conceptual simplicity was a major consideration in the early search for accurate methods. Much time and effort went into testing new types of wavefunctions such as GVB,,O strongly orthogonal geminals,21 valence-space MCSCF2 etc. These wavefunctions were regarded as conceptually simple extensions of SCF wave function^.^^ Direct calculation of configuration interaction wavef~nctions~~ with non-orthogonal basis functions was abandoned because it was feasible only for a few electrons and also because it led to wavefunctions which could not be interpreted.At present configuration interaction with configurations built from orthonormal orbitals as advocated by Boys,25 has been adopted as the only tractable open-ended scheme.26.27 Use of MCSFC28 or natural orbitalsl9$ 29 in the CI reduces the number of configurations needed but this method is still criticized for the lack of conceptual simplicity of the resultant wavefunctions. This criticism is very much to the point when an elaborate calculation is carried out as I recently did for a molecule such as water at its ground-state equilibrium geometry.30 Most chemists will concede that the Schrodinger equation is correct and that the exact solution will reproduce the expeximental binding energy dipole moment etc.Hence a calculation must do more than merely agree with the experimental facts without offering an interpretation of why the dipole moment or binding energy has that particular value. On the other hand correlation effects are not simple. It is in fact a contradiction to ask for a wavefunction which is both simple and accurate. Calibration tests of computational schemes on molecules like water are important to establish credibility of a new method before it is applied to molecules where the experimental facts are in question. In doing elaborate calculations of properties of molecules however it E. R. DAVIDSON must always be kept in mind that the experimentalist who measures these properties does so in order to learn about the electronic structure of the molecule.A calculation should not only reproduce the experimental properties but it should also offer the correct interpretation of why the property has that particular value. ATOMIC ORBITALS The new model of the chemical bond which developed during the 1960s was a logical extension of the LCAO-MO-SCF model which had previously failed to provide quantitative results. The new model is still built at least conceptually on near Hartree-Fock atomic orbitals. In order to obtain quantitative results it is essential to add to this basis additional functions which can account for the change in size and shape of the valence orbitals in the molecular environment.Hence the simplest basis set for a semiquantitative model is the split-valence plus polarization basis introduced by Ne~bet.~l Chemists who are used to thinking in terms of loosely defined 'atomic orbitals' find it hard to interpret the results based on this extended basis set. The fact that d orbitals are essential to get the correct bond angles in hydrogen peroxide3' or the correct bond length in oxygen33 goes beyond the crude model based on unmodified atomic orbitals. Because integrals are easier for Gaussian basis functions these orbitals are usually approximated as linear combinations of Cartesian Gaussian orbitals. For computa- tional simplicity a linear transformation of this basis is often made to a new basis spanning the same vector space but requiring fewer Gaussians per basis function.Also core orbitals are sometimes represented in lower accuracy in the hope that any errors introduced will cancel in computing energy differences. This contracted-orbital approach was pioneered by Whitten34 and is used in almost all molecule calculations. Split-valence plus polarization Gaussian basis sets were deveioped independently by P~ple~~ and MOLECULAR ORBITALS In the model of 1970 molecular orbitals were constructed from the atomic basis by solving SCF equations for the electronic state of intere~t.'~ The virtual orbitals from SCF calculations proved useless for describing electron correlation so they were transformed to natural orbitals by an iterative process.lg A different set of molecular orbitals was used for each state of the molecule.During the decade following 1970 some refinements of the method for constructing molecular orbitals was made. The GVB method of Goddard2* and Wahl's limited MCSCF calculations2s were expanded to include more general MCSCF wavefunctions as methodological improvements (and faster computers) made this feasible. These more general wavefunctions incorporate the virtues of valence-bond wavefunctions without the computational difficulties of dealing with non-orthogonal basis functions. They still require different orbitals for different states and some modification of virtual orbitals to make them useful for describing electron c~rrelation.~~ These generalized molecular orbitals do not give orbital energies which can be used to form a molecular-orbital energy diagram of the type popular in inorganic texts.The interpretation of the orbitals used to describe electron correlation is difficult because there is usually no electronic state of the molecule in which these orbitals play a dominant role. GENERAL INTRODUCTION WAVEFUNCTIONS In 1970 the correlation effect on the energy and properties was estimated by carrying out a variational calculation using the double excitations which enter the wavefunction in first-order perturbation theory. Single excitations were also included because they affect properties other than the energy in the same order of perturbation theory as the double excitation^.^^ Peyerimhoff Bender and Schaeffer to name a few investigated the ground and excited states of several m01ecules.~~ Following 1970 configuration interaction was enlarged to include a more general approximation based on including all configurations with large coefficients as part of the zeroth-order wavefunction plus all configurations mixing with these in first order.26 Recognition of the fundamental inability of CI to describe correlation effects in large led to an increased interest in perturbation theory.Unfortunately the simplest form of perturbation theory proved inadequate because it could not handle zeroth-order wavefunctions requiring several configuration^.^^ However perturbation theory showed that triple and quadruple excitations were more important than previous CI calculations had assumed.The standard ‘state-of-the-art ’ ab initio model of 1980 certainly includes all single and double excitations from all the important zeroth-order configurations. Configuration interaction and perturbation theory are merging into a more unified approach based on linked-cluster experiments unlinked-cluster corrections and variation-perturbation theory. Accurate wavefunctions are complicated. Fortunately one can usually understand the zeroth-order wavefunction. The other terms merely describe the fact that electrons avoid each other and are needed for quantitative but not qualitative results. Use of charge densities bond orders etc. to interpret complex wavefunctions greatly facilitates understanding them. 42 APPLICATIONS Many calibrated ab initio packages such as GAUSSIAN 80 are now available which allow any chemist access to the predictions of this modern model of the chemical bond.Interest in the results has been spurred by the technological advances which make geometry optimization easier. As an outgrowth of Pulay’s ideas,43 optimization based on analytical gradients is now routine for both SCF and MCSCF calculations. One application of these methods to electronic structure has been the prediction of the singlet-triplet energy difference of methylene. Over the years both theory and experiment have given a wide variety of results for this molecule. Since 1970 however the theoretical results have agreed on the bond angles and relative energies.44 Table 3 shows that the energy difference does converge with improved calculations.As in the previous examples atomic orbitals give the wrong results but with allowance for distortion in size and shape the correct result is easily obtained. Because the correct result differed from some experiments much more elaborate calculations have been done to verify the convergence. The ability to compute derivatives of the potential energy has made it possible to enlarge the studies of molecular spectra to include vibrational frequencies and intensities in addition to electronic excitation energies. Some examples of this are presented in this Symposium by Peyerimh~ff~~ and S~haeffer.~~ The ideas introduced by Woodward and Hoffmann4’ to explain chemical reactivity have also popularized calculations among organic chemists.The ability to predict the transition-state geometry and to follow the Fuk~i~~ reaction path features not accessible to experiment has increased interest in the theoretical results. A typical E. R. DAVIDSON Table 3. Convergence of the singlet-triplet energy difference of methylene (in kJ mol-l)a* basis SCF TCSCF CI minimum 167 139 DZ 134 96 100 DZP 109 54 63 extended 105 46 46 a Double-zeta basis (DZ) allows for change in size of atomic orbitals polarization (DZP) allows also for change in shape. Extended basis sets allow for better description of these effects as well as better description of electron correlation. The TCSCF results are based on using a single spin-restricted SCF wavefunction for the triplet and a two-configuration description for the singlet.Two configurations are needed for the singlet in a zeroth-order description at larger bond angles. example of reactions which have been studied is the methylenecyclopane rearrangement.49 This molecule may undergo either stepwise or concerted ring opening to trimethylene methane. Calculations located three transition states and agreed with experiment that the non-concerted path was lowest. Another reaction illustrating this same question of step-wise versus concerted reactions is the decomposition of 2-carbena- 1,3-dioxolane to ethylene and carbon dioxide. Even though the concerted reaction is Woodward-Hoffmann allowed calculations indicate the reaction proceeds step-wise with a diradical intermediate.50 The paper by Bernardi et al.in this Symposium deals with a similar reaction.51 More recently the Cope rearrangement has been studied. In this case the calculations indicate the reaction is concerted and passes through a symmetrical transition statel’ rather than proceeding via a diradical intermediate. Now that an adequate model exists for predicting the electronic structure of small gaseous organic molecules over a wide range of geometries and excitation levels one might wonder whether there are problems left requiring a truly ab initio approach. The answer is definitely yes. Calculations on metallo-organic and metal-metal bonds are still unsatisfactory and not r0utine.~~9 53 Little progress has been made on studying reactions in s01ution.~~9 Properties like the spin density56 and field gradient30 are 55 inadequately predicted by standard basis sets.The standard model has not even been applied to such a simple problem as predicting the structure of NaCl (solid) so it is uncertain whether it will predict the correct crystal structure. Calculations on the structure of ice indicate that it is very difficult to predict the correct 0-0 and j8 0-H distances and proton field gradients in a hydrogen-bonded On a more mundane level it should be noted that even the electron affinity of carbon and oxygen atoms are difficult to compute to better than 10 kJ mol-1 even with expanded basis sets and CI.59Thus there remains ample challenge to those wishing to do ab initio calculations. At the same time we can look with satisfaction at the very useful model which has developed over the past 20 years.This model should be introduced into freshman texts and other descriptive works as a refinement of the common LCAO-MO-SCF and VB models. The model itself is actually fairly simple even though a computer is required to work out the implication of the model for any particular molecule. 14 GENERAL INTRODUCTION G. H. Hardy A Mathematician’s Apology (Cambridge University Press Cambridge 1969). C. A. Coulson Rev. Mod. Phys. 1960 33 170. a J. 0.Hirschfelder Annu. Rev. Phys. Chem. 1983 34 1. L. Pauling The Nature of the Chemical Bond (Cornell University Press Ithaca NY 1940). Y. Tanabe and S. Sugano J. Phys. Soc. Jpn 1954 9 753. R. S. Mulliken Phys.Reti. 1936 50 1017. E. Huckel Z. Phys. 1931 70 204. A. D. Walsh J. Chem. Soc. 1953 2260. R. Hoffmann J. Chem. Phys. 1963 39 1397. ” J. A. Pople and G. A. Segal J. Chem. Phys. 1966 44 3289. M. J. S. Dewar and W. Thiel J. Am. Chem. Soc. 1977,99 4899. l2 E. Schrodinger Ann. Phys. 1926 79 361. l3 W. Heitler and F. London 2. Phys. 1927,44 455. l4 D. R. Hartree Proc. Cambridge Phil. Soc. 1928 24 89. l5 E. Hylleraas Z. Phys. 1930 65 209. l6 H. M. James and A. S. Coolidge J. Chem. Phys. 1933 1 825. l7 Y. Osamura S. Kato K. Morokuma D. Feller E. R. Davidson and W. T. Borden J. Am. Chem. Soc. 1984 106 3362. l8 S. Rothenberg and E. R. Davidson J. Chem. Phys. 1966 45 2560. l9 C. Bender and E. R. Davidson J. Phys. Chem. 1966,70 2675. 2o W. J. Hunt P.J. Hay and W. A. Goddard 111 J. Chem. Phys. 1972 57 738. *l A. C. Hurley J. E. Lennard-Jones and J. A. Pople Proc. R. Soc. London Ser. A 1953 220,446. 22 G. Das and A. C. Wahl J. Chem. Phys. 1966,44 87. 23 C. C. J. Roothaan Rev. Mod. Phys. 1951 23 69. 24 J. C. Browne and F. A. Matsen Phys. Rev. 1964 135 A1227. 25 S. F. Boys Proc. R. Soc. London Ser. A 1950 200 542. 26 R. J. Buenker S. D. Peyerimhoff and W. Butocher Mol. Phys. 1978,35 771. 27 I. Shavitt in Methods of Electronic Structure Theory ed. H. F. Schaeffer I11 (Plenum Press New York 1977). 28 A. C.Wahl and G. Das in Methods of Electronic Structure Theory ed. H. F. Schaeffer I11 (Plenum Press New York 1977). 29 P. 0.Lowdin Phys. Rev. 1955 97 1474. 30 D. Feller and E. R. Davidson Chem. Phys.Lett. 1984 104 54. 31 R. K. Nesbet Phys. Rev. 1968 175 2. 32 T. H. Dunning and N. W. Winter Chem. Phys. Lett. 1971 11 194. 33 R. A. Whiteside M. J. Frisch J. S. Binkley D. J. DeFrees H. B. Schlegel K. Raghawahri and J. A. Pople Carnegie-Mellon Quantum Chemistry Archive (Carnegie-Mellon University Pittsburgh PA 2 vol edn 1981). 34 J. L. Whitten J. Chem. Phys. 1963 39 349. 35 P. C. Hariharan and J. A. Pople Theor. Chim. Acta 1973 28 213. 36 T. H. Dunning Jr and P. J. Hay in Methods of Electronic Structure Theory ed. H. F. Schaeffer I11 (Plenum Press New York 1977). 37 D. Feller and E. R. Davidson J. Chem. Phys. 1981 74 3977. 38 E. R. Davidson Reduced Density Matrices in Quantum Chemistry (Academic Press New York 1976). 39 E. R. Davidson and L.E. McMurchie in Excited States ed. E. C. Lim (Academic Press New York 1982) vol. 5. 40 E. R. Davidson and D. W. Silver Chem. Phys. Lett. 1977 52 403. 41 R. J. Bartlett Annu. Rev. Phys. Chem. 1981 32 359. 42 E. R. Davidson J. Chem. Phys. 1967 46 3319. 43 P. Pulay Mol. Phys. 1969 17 197. 44 E. R. Davidson in Diradicals ed. W. T. Borden (Wiley New York 1982). 45 S. D. Peyerimhoff Faraday Symp. Chem. Soc. 1984 19 63. 46 M. E. Colvin and H. F. Schaeffer 111 Faraday Symp. Chem. Soc. 1984 19 39. 47 R. B. Woodward and R. Hoffmann The Conservation of Orbital Symmetry (Academic Press New York 1971). 48 K. Fukui J. Phys. Chem. 1970 74 4161. 49 D. Feller K. Tanaka E. R. Davidson and W. T. Borden J. Am. Chem. Soc. 1982 104 967. 50 D. Feller E. R. Davidson and W.T. Borden J. Am. Chem. Soc. 1981 103 2558. j1 F. Bernardi A. Bottoni J. J. W. McDouall M. A. Robb and H. B. Schlegel Faraday Symp. Chem. Soc. 1984 19 137. A. D. McLean and B. Liu Chem. Phys. Lett. 1983 101 144. j3 P. E. M. Siegbahn Faraday Symp. Chem. Soc. 1984 19,97. E. R. DAVIDSON 54 J. Chandrasekahn S. F. Smith and W. L. Jorgensen J. Am. Chem. Soc. 1984 106 3049. 55 K. Morokuma K. Ohta N. Koga S. Obara and E. R. Davidson Faraday Symp. Chem. SOC.,1984. 19 49. 56 D. Feller and E. R. Davidson J. Chem. Phys. 1984 80 1006. 57 E. R. Davidson and K. Morokuma J. Chem. Phys. 1984,81 3741. 58 E. R. Davidson and K. Morokuma Chem. Phys. Lett. 1984 111 7. 59 D. Feller L. E. McMurchie W. T. Borden and E. R. Davidson J. Chem. Phys. 1982 77 6134.

ISSN:0301-5696    

DOI:10.1039/FS9841900007

出版商:RSC    

年代:1984

数据来源: RSC

摘要:

Faraday Symp. Chem. SOC.,1984 19 17-37 The Lennard-Jones Lecture The Value of Very-high-accuracy Calculations in Quantum Chemistry By N. C. HANDY University Chemical Laboratory Lensfield Road Cambridge CB2 1EW Receiued 14th January 1985 Five recent topics under research at Cambridge in ab initio quantum chemistry are reported (a) the latest developments in methods for full CI calculations (b) the convergence of the Marller-Plesset perturbation-theory series at the RHF and UHF level using H,O and NH as examples with energies up to 24th order (c) a new way to calculate Marller-Plesset energy gradients and second derivatives the latter needing only the solution of the first-order CPHF equations (d) some calculations using the quantum Monte Carlo method compared with standard CI calculations and (e) high-accuracy calculations on Be, Be and Be to demonstrate the value of such research.I. INTRODUCTION This paper reviews some of the progress made in the development of methods for the accurate ab initio computation of the properties of molecules by members of the Cambridge University Theoretical Chemistry Department. Some particular applica- tions of these methods are also reported. The members of the department who have been actively involved in this research are Drs R. D. Amos S. M. Colwell R. J. Harrison P. J. Knowles and H-J. Werner and Miss J. E. Rice and Miss K. Somasundram. Much stimulation for this research came from interaction with Prof. H. F. Schaefer’s group at Berkeley and as far as these particular topics are concerned helpful conversations were held with Prof.J. A. Pople R. Ahlrichs P. Siegbahn and B. Alder. In our research we must take care to keep up with and take advantage of the latest hardware advances in modern computer technology. It will be seen in the forthcoming sections that considerable emphasis is placed on the importance of developing methods which take advantage of the ‘vector’ capability of the CRAY-IS computer to which we have had access over the past three years. (In my experience there is always at least a year’s development time before researchers think in the appropriate way to take such advantages.) We also imagine that in the next few years there will be considerable emphasis on the development of programs for the smallest computers (e.g.IBM PC) which for a few thousand dollars can give the C.P.U. power of a VAXl 1/750. Since I have spent much of my research on the development of methods for highly accurate wavefunctions this paper will be largely devoted to such topics. These are now summarised. (a) As a test of the accuracy of many methods it is desirable to have an efficient program for full configuration-interaction (CI) calculations. Much progress has been made on this problem in recent years and in section I1 a new method for these calculations is introduced which takes advantage of vectorisation procedures. Besides being useful in its own right this program can be the central algorithm of any complete active-space self-consistent-field (CASSCF) program.17 THE LENNARD-JONES LECTURE (b) The wide usage of the GAUSSIAN-80 and GAUSSIAN-82 packages and their reliance on Marller-Plesset perturbation theory for the calculation of correlation energies forces one to enquire of the convergence of this perturbation series. Using the full CI program it is possible to examine the convergence for some selected cases and these new results are discussed in section IV. In this context the accuracy of single-plus-double excitation CI(SDCI) many-body perturbation theory (MBPT) and multi-reference SDCI (MRSDCI) calculations in general are also discussed in this section. (c) We have recently observed that the way in which energy gradients and higher derivatives are evaluated for energies which include electron correlation can be improved.In section V it is shown that second-order Marller-Plesset second derivatives can be evaluated without solving the 3N2 second-order CPHF equations (N is the number of nuclei). (d) The recent innovations in the area of quantum Monte Carlo calculations for the electronic energies of small systems has led to some claims that this approach may be the answer to many of our problems. In section VI this method is compared with more standard ab initio methods and future areas for development of the method are discussed. (e) Finally as a demonstration of the very-high-accuracy type of calculations that we are now able to perform results are reported for the binding energies of Be, Be and Be,. Theoreticians predicted the existence of a bound 'Cg state of Be with several vibrational levels and this has now been confirmed experimentally.11. THE FULL-CI PROCEDURE This is a very simple problem to state given a set of orthonormal orbitals 41&...4m what are the energies obtained for the lowest states when the wavefunction is expressed as a linear combination of all possible determinants which can be formed from these orbitals? The length of the expansion is reduced if configuration state functions (CSF) are used instead of determinants but the results will be the same. Full CI calculations involve differently structured programs to the usual variational programs because all the two-electron integrals may be held in core. However the length of the CI vector rapidly becomes very long.If A4 is the number of orbitals and N and Npare the number of aand /3 electrons the number of determinants is When the Sz eigenvalue is zero it is possible to show that the combinations &(@% @4J & &(@BA @%) (2) span singlet and triplet states respectively. In this notation OAand denote a string of orbitals. The following table shows approximate relationships between the number of CSF and the number of determinants which arise in typical cases szeigenvalue state expansion function no. of determinants/ no. of CSF 4 1 2 1 1.2 1 2.6 1 1.8 1 1.4 1 1.2 1 N. C. HANDY In the full-CI programs we have developed we have used determinants as the expansion functions. If the combinations (2) are used this means that the length of the determinant vector will be no more than twice as long as the CSF vector except for the doublet case.This probably means that as far as full-CI is concerned the use of determinants is not a major disadvantage. The first full-CI program which was designed for the largest cases used the combinations (2)1-3and used the Cooper-Nesbet algorithm4 to determine the lowest eigenvector. This relies upon Acz = a1/(Hzz-E) (3) as the update procedure performed sequentially. It needs only the current eigenvector in the memory. The program was very short easy to debug and has now been used by various groups. The program is not totally vectorisable because it needs random access to the integral list and is fairly slow. It relies upon a lexical ordering of the determinants; this is possible because the orbital occupancy can give the /Iand a string labels which then give the determinant label.The largest case we have considered is a calculation on HF involving 1880346 combinations,2 or 944348 CSF in C, symmetry. This is probably the best program available for large full-CI calculations. It is possible to extend the program when it is not possible to hold even the one vector in the memory by the standard ‘paging’ techniques but this makes the program slower. In principal therefore there is no limit to the size of calculation this program can perform. In a recent publication5 Siegbahn showed how it is possible to vectorise completely the full-CI algorithm. The method relies on the Davidson algorithm6 and therefore needs two vectors in the memory.For maximum efficiency paging should not be considered. Essentially his idea was to write the evaluation of the cvector as a matrix product or where the coupling coefficients r& can be written in terms of the unitary group one-particle generators :7 which may be written as r;;Ll= k x(11 Eij I K)(Kl Ekl I J>-$(I1Ei I J> (7) K on the introduction of the resolution of the identity. The difficult term is the first and its contribution to oImay be written as This formulation leads to the scheme 0= Tr(y I-D) THE LENNARD-JONES LECTURE kl ij which can be efficiently vectorised with the sum over K as the outermost loop provided the one-electron coupling coefficients yGK (given K all J) are available from a tape or can be rapidly evaluated.Siegbahn based his program on the graphical unitary group approach (GUGA) CI algorithm,8-10 which meant that the ygK had to be predetermined and held on a tape. This meant that the method rapidly became input- output bound. The largest case Siegbahn reported was one involving 30000 CSF. If instead of CSF single determinants are used the evaluation of the ygK is trivial; they each have value 0 or 1. We have recently reported the development of this determinant CI algorithm.ll The determinants are addressed cia a lexical ordering scheme which means that the whole procedure is vectorisable on the CRAY and the algorithm is one of the most ‘vector’4ntensive in existence. Full details are given in ref.(1 1). Because of the structure of the Davidson iterative algorithm if one starts with a guessed vector which has the correct space-spin symmetry (ie.a linear combination of a small number of determinants) then that symmetry will be exactly maintained throughout the procedure provided that the diagonal elements HI are replaced by the average diagonal energy l/MZyl HI,of all those A4 determinants Orwhich have the same orbitals singly occupied. This aspect is now fully vectorised as well. On the CRAY-IS with 106 words of memory it is possible to consider calculations with 300000 determinants d(@i @&) and for such a case an iteration takes 18 C.P.U.s with no input-output requirements. This program is now available from the authors.An important by-product of this algorithm is that it forms the central part of a new CASSCF12 program developed by Werner and Knowles.13 This latest multiconfigur- ation SCF (MCSCF) development has examined carefully the coupling between orbital and CT coefficients and the evidence to date is that the new program has excellent convergence features and can deal with systems involving > 100000 CSF which means that molecules involving transition-metal atoms can be reasonably treated calculations on Fe0(5A) are being reported. The conclusion of this section is that the last five years have seen the development of highly efficient vectorised codes for full-CI calculations which besides being useful on their own for theoretical investigations on convergence form the central part of good CASSCF programs.It should also be added at this stage that the GUGA CI program is suitable for full-CI calculations when the number of electrons is very small and the number of basis functions is large. Such a calculation is not really appropriate for GUGA but we have reported calculations on Be and (H,).l* 111. THE GENERATION OF TERMS IN THE M0LLER-PLESSET PERTURBATION SERIES We may use the full-CI program previously described to examine the convergence of the Msller-Plesset perturbation series.I5 In the framework when the zeroth-order hamiltonian is the sum of one-electron Fock operators and we use standard Rayleigh-Schrodinger perturbation theory then many-body perturbation theory (MBPT)l67l7and Mdler-Plesset (MP) theory involve the same series the names only distinguishing the way in which the early terms in the series are evaluated.For N. C. HANDY higher-order terms there seems only one sensible way to proceed and that is outlined by Bartlett and Brandasla and reproduced here. Ho is defined as Ho = xF(i). (14) The eigenfunctions of Hoare all the determinants OIformed from the set of all orbitals involved and the eigenvalues are the sum of the orbital energies. Intermediate normalisation is used (yoI Vk) = 0 and the perturbation equations are Ho Yo = Eo Yo (15) k (HO-EO) lyk+(H1-El) wk-1-c Er Vk-r = (16) where H = Ho+HI. The full-CI program evaluates vectors o = Hc and thus in this case it can evaluate Ok = Hvk-l (17) where the ly are expanded in terms of all the determinants V/k = xdrk(Dr* (18) Successive perturbation energies can therefore be evaluated through Ek = (YO I Ok-1) (19) and i.e.to evaluate the kth perturbation energy needs a time equivalent to k full-CI iterations. Thus this development is a trivial change to the full-CI program for closed-shell restricted-Hartree-Fock (RHF) calculations. Of course MP theory may also be used in the unrestricted-Hartree-Fock (UHF) frame~0rk.l~ The full-CI program is amended to have available the two-electron integrals (aaI aa) (aaPP) (JP I aa) and (apI PP) in the memory. There are no other changes necessary to enable the program to generate nth order MP UHF energies. Laidig et uf.,O have also reported MBPT(n) series calculations based on their GUGA programs.IV. RESULTS AND CONCLUSIONS FROM THE FULL-CI AND PERTURBATION SERIES CALCULATIONS Several full-CI calculations have now been performed on systems with four or more electrons BH (2 -[+polarisation basis) H,O (2 -< basis) NH (2 -[basis frozen core and virtual orbital) HF (2 -[+polarisation basis frozen core and virtual orbital),,T and some ionised states (,B, 2Al and ,B,) and excited states (,B1 ,B2 ,A1 and ,A,) of H,0.21 Whilst it is obviously important to have many such calculations available in particular with as many basis functions and electrons as possible so that there are many types of higher excitations this author believes that probably many of the conclusions which can be deduced from a careful examination of the results for H,O (lA,) at three C, geometries (6 = 8, r = re l.5re and 2.0rJ can be extrapolated to many other systems.The significance of these calculations is that at the full-CI level the coefficient THE LENNARD-JONES LECTURE Table 1. Calculations on H,O using a 2-[ basis set. All orbitals active. Three bond lengths. C, symmetry ~ ~~ re 1.5re 2.0re SCF" -76.009 838 -75.803 529 -75.595 180 SDCI" -76.150 015 -75.992 140 -75.844 817 (0.9 78 7) (0.9462) (0.8714) SDTQCI" -76.157 603 -76.013 418 -75.900 896 (0.9 754) (0.9247) (0.7726) full CI" -76.157 866 -76.014 521 -75.905 247 (0.9 7 5 3) (0.923 3) (0.7636) Davidson extrapolationa -76.155 912 -76.01 1 867 -75.904 874 MBPT(2)b -76.149 315 -75.994 577 -75.852 460 MBPT( 3)b -76.150 707 -75.989 39 1 -75.834 803 MBPT(4)b -76.156 876 -76.008 395 -75.888 867 M BPT( 5)" -76.157 056 -76.009 771 -75.889 199 [2 11" -76.157 358 -76.012 555 -75.905 780 CCSDb -76.156 076 -76.008 931 -75.895 913 CCSD +T(4)b -76.157 438 -76.012 858 -75.907 996 CASSCF-6C -76.026 113 -75.842 122 -75.677 112 CASSCF-7 -76.065 100 -75.927 258 -75.828 416 CASSCF-8 -76.099 399 -75.955 930 -75.845 671 CASSCF-9 -76.132 645 -75.982 052 -75.866 155 MRCISD-6" -76.152 546 -76.004 199 -75.870 295 MRCISD-7 -76.155 929 -76.012 464 -75.903 349 MRCISD-8 -76.157 488 -76.014 032 -75.904 708 MRCISD-9 -76.157 783 -76.014 349 -75.905 105 " From Handy and coworkers ref.(1)-(3). From Bartlett and coworkers ref. (17) and (20). From Shavitt and coworkers ref.(22). of the dominant determinant is 0.975 0.923 and 0.763 respectively and thus such calculations will show clearly the effects of configuration mixing which are met in so many calculations especially away from equilibrium geometry. The size of the calculations was 502355 determinant combinations (2) or 256473 CSF. In table 1 many results are collated by various methods using the same basis set and geometries. We now discuss the results under the various headings in that table. Acknowledgement is made here to Prof. Bartlett and Shavitt whose results are q~0ted.l~~ 20* 22 SCF (self-consistent field). Note that at re l.5re and 2.0r the correlation energies are 0.148028 0.210992 and 0.310067 hartree respectively. SDCI. The singles and doubles CI calculation picks up 95 89 and 80% of the correlation energy.This is clearly an inferior method for potential-energy surfaces. SDTQCI. With triples and quadruples added although all results are better than 98.5% of the correlation energy the error at 2re is 0.00435 hartree. 17678 CSF are involved. Davidson correction This extrapolation gives the best result for 2re but must be regarded as fortuitous! MBPT(4). This is the limit of general practical calculations of MP or MBPT. The errors are 0.000990 0.006 126 and 0.01638 hartree. These results demonstrate the weakness of this method for sections of surfaces involving the breaking of bonds. N. C. HANDY 0.012 0.013 0.014 0.015 0.016 0.017 0.0 18 0.019 11111111111 3 4 5 6 7 8 9 1011 121314 Fig.1. Convergence of MP-n series for H,O at Re:SCF =-75.8884 2nd order = -76.0093 exact = -76.01851 5. CCSD. The coupled-cluster SD calculations have errors of 0.00179 0.00559 and 0.009334 hartree; the errors are comparable to MBPT(4). CASSCF. Even with 9 active orbitals only 83% of the correlation energy at Y is obtained. MRCISD. These SDCI calculations using the CASSCF functions as the reference set are by far the most uniformly accurate. Even SD-CI out of CASSCF with 7 active orbitals shows a constant error of 0.001 hartree at the three points (7096 CSF in these calculations). Some of these results are given in fig. 1 which shows very clearly that the best results are achieved by MRCISD calculations; and the conclusion of this discussion must be that the very best programs are going to be those which involve SD excitations from a carefully selected CASSCF or MCSCF reference space.The most hopeful attempt in this direction to date is due to Werner and Me~er~~ in their contracted CI scheme where SD replacements of the full reference function are included in a CI scheme. This is where the future may lie. The results of the perturbation-theory calculations which are given in table 2 and THE LENNARD-JONES LECTURE Table 2. Marller-Plesset perturbation energies En using 6-21G basis set with a frozen core orbital for re lSre and 2.0re(8 = 107.6') for H20 En re 1.5re 2.0re -0.120 865 -0.166 896 -0.241 643 -0.003 303 -0.002 015 0.006 123 -0.004 849 -0.016 925 -0.046 465 -0.000 488 -0.001 352 -0.004 219 -0.000 435 -0.003 417 -0.012 369 -0.000 076 -0.000 338 -0.001 032 -0.000 048 -0.000 827 -0.004 065 -0.000 012 -0.000 032 0.002 319 -0.000 006 -0.000 189 -0.001 417 -0.000 002 0.000 010 0.002 191 -0.000 00 1 -0.000 032 -0.000 277 10-7 0.000 007 0.000 959 10-7 -0.000 001 0.000 023 10-7 0.000 003 0.000 146 10-8 0.000 002 0.000 025 10-8 0.000 001 -0.000 140 10-9 0.000 001 0.000 043 10-9 10-7 -0.000 159 10-9 1o-6 0.000 087 E21 10-10 10-7 -0.000 095 10-10 10-7 0.000 095 E22 E23 1O-IO 10-8 -0.000 037 10-l1 lo-* 0.000 064 E24 fig.2-4 show that in most cases the series will converge to the exact eigenvalue. The calculations we have performed on H20used a 6-21G basis with a frozen core orbital the latter being to make the size of calculation tractable (61441 determinants and 245025 internal IK) determinants).Indeed the usual applications of MP theory using GAUSSIAN~~~ 25 programs use frozen core orbitals. That the effect of the core orbital does not affect a general discussion of the convergence of the series is indicated from a comparison of the MP-n (n = 2-5) perturbation energies En for the 24 (no frozen core) and these 6-21G calculations the former being due to Laidig et al. for H,O at 2r 2-r 6-21G E2 -0.257 28 -0.241 64 E3 0.017 66 0.006 12 E4 -0.054 06 -0.046 47 -0.000 33 -0.004 22 E5 -0.012 37 E6 The results in table 2 clearly show that the early rapid convergence of the MP-n series is not maintained.In the most favourable case re the energies have converged to within 0.001 hartree with E4,whilst for l.5re this happens at E, and for 2r, at 0.89 0.894 \ 0.895 0.896 ! I I 0.89 7 X 0.898 \ X \ 0.899 0.900 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 Fig. 2. Convergence of MP-n series for H,O at 1.5Re SCF = -75.7072 2nd order = -75.8741 3rd order = -75.8761 exact = -75.899202. 0.789 0.790 0 -791 0.792 0.793 0.794 v, 0.795 -X I I ,I , I ,, I Fig. 3. Convergence of MP-n series for H,O at 2.0Re SCF = -75.4914 2nd order = -75.7730 3rd order = -75.269 t3 4th order =-75.7733 5th order = -75.7776 exact = cn -75.791 269. THE LENNARD-JONES LECTURE 0.01 I 0.009 -0.008 -0.007 -0.006 -0.005 -0.OOL -0.003 -0.002 -0.001 0.001 0.002 0.003r 0.OOL 0.005 Fig.4. Energies of H20(2-c basis) using various methods (a) SDCI (b) MBPT(4) (c) MBPT(5) (d)SD T QCI (e)MRCISD-7 and (f)MBPT(8). Table 3. SCF eigenvalues of the 6-2 1G H,O calculations En re 1Sr 2.0re E2 -1.329 -1.213 -1.199 E3 -0.684 -0.506 -0.504 E4 -0.529 -0.484 -0.415 E5 -0.475 -0.465 -0.409 ‘6 0.261 0.099 -0.012 E7 0.361 0.170 0.033 ‘8 1.207 1.081 1.121 E13.The fact that the convergence is so rapid at re underlines the value of the many MP-4 calculations which are carried out at equilibrium geometries. However away from equlibrium the picture changes. Convergence is reasonable at 1.5re but note that at E5the error is 0.005 hartree which means that it is impractical to achieve 0.001 hartree accuracy.At 2re at eighth order the energy is 0.0038 below the exact value whilst at 22nd order the energy is oscillating by 0.0001. These results at 2re show the dangers of using any Pade approximants.18 These approximants are designed to approximate the energy that will be obtained with a few more terms in the series. For example an approximant based on E3 E4and E5 yields an energy -75.792 704 which is 0.0013 below the exact. The [3,3] approximant yields -75.7993 which is 0.008 below the exact. In this author’s view any approximant which is claimed to be successful at these distorted geometries can only be so by chance. A reason for the slower convergence at 1.5re and 2re can be obtained by observing the SCF eigenvalues in table 3.The two lowest virtual eigenvalues E and E, become N. C. HANDY Table 4. UHF MP-n calculations on NH (,B,) using 6-31G basis (frozen core) re 13 2re -55.530 176 -55.367 729 -55.181 -55.532 247 -55.405 143 -55.381 93 1 -0.085 5 12 -0.062 116 -0.031 541 -0.009 815 -0.01 1 393 -0.006 208 -0.003 612 -0.008 695 -0.002 508 -0.001 192 -0.005 775 -0.001 235 -0.000 463 -0.004 888 -0.000 845 -0.000 200 -0.004 066 -0.000 663 E* -0.000 100 -0.003 505 -0.000 568 CE -55.633 142 -55.505 582 -55.425 497 full CI -55.633 27 -55.526 -55.44 correlation energy RHF 0.103 0.158 0.26 UHF 0.101 0.121 0.06 (9) 0.752 1.661 2.51 0.627 0.628 0.629 0.630 0.631 I 0.632 0.633 I \ 2345678 Fig.5. Convergence of MP-n series for NH at Re:SCF = -55.53225,2nd order =-55.61 776 exact = -55.63327. THE LENNARD-JONES LECTURE 0.487 0.489 0.491 0.493 0.495 0.497 0.499 0.501 0.503 0.505 0.507 0.509 0.51 1 0.512 I I I I 1 I I 3 4 5 6 7 8 9 1 0 Fig. 6. Convergence of MP-n series for NH at 1.5Re SCF = -55.40514 2nd order = -55.46726 3rd order = -55.47865 exact = -55.526. nearly degenerate and indeed E < 0 at 2re. The MP series is so dependent on the size of the denominators that this has to be the principal reason for the convergence difficulties. We deduce that MP-n [or MBPT(n)] calculations ought to be viewed with serious suspicion if any of the occupied or virtual orbitals have the wrong sign for their SCF energies.We also have some preliminary results for UHF MP-n theory. The calculations were performed for NH (2B,),for fixed bond angle 8 = 103.2" and bond lengths re 1.5re and 2re where re = 1.013 A (C2vsymmetry) with a 63 1G basis set and a frozen core orbital. The results are given in table 4 and fig. 5-7. The results are rather different to those for the closed-shell RHF case. In all three cases the convergence appears smooth but it is very slowly convergent at l.5re and 2re. It appears to be sufficiently smooth that some sort of extrapolation may be useful this is being investigated. The results may be considered curious because although UHF picks up 2 23 and 77% of the correlation energy at re 1.5re and 2re the series needs many more terms to find the remaining correlation energy at 2re than at re.The reason of course N. C. HANDY 0.420 \ 0.422 0.423 0.424 0.425 0.426 3 4 5 6 7 8 9 10 Fig. 7. Convergence of MP-n series for NH at 2.0Re SCF = -55.38193 2nd order = -55.41347 exact = -55.439. lies in the very poor spin state found by the UHF calculation as is seen by (S2) = 2.5 at 2r,. In summary preliminary conclusions on these investigations into the convergence of MP-n series are that (a) the RHF results show slow and irregular convergence features away from equilibrium and (b) the UHF results show a much smoother convergence but also a very slow convergence away from equilibrium.Further investigations will be carried out and a complete set of results and discussion given elsewhere. V. AN IMPROVED THEORY FOR THE CALCULATION OF ENERGY GRADIENTS AND HIGHER DERIVATIVES Because the calculation of energy gradients and second derivatives is becoming an important every day tool for the ab initio chemist it is useful at this stage to underline a recent development in the theory for these calculations which has particular relevance for the calculation of CI gradients and MP-2 first and second derivatives. The possibility of calculating MP-2 second derivatives analytically is important 30 THE LENNARD-JONES LECTURE .~~ because as Pople et ~1say 'MP-2 accounts for 70% of the difference between HF theory and experiment yielding frequencies which differ from experiment by only 10-90 cm-l '.Under a movement of a nucleus the orbitals and the basis-function integrals 28 4i 4 4i+a CUai4a (21) + (aBI Y@ CaB I 74+4og I 74' (22) and for SCF or MCSCF wavefunctions the energy gradient can be straightforwardly evaluated29 with the usual notations for the integrals and the basis-function density matrices y r. In the derivation of eqn (23) the uUican be eliminated because the energy E has been optimised with respect to all parameters. For the MP-2 energy26 occ virt EM, =-a z E (iJ' II +ESCF ij ab (EafEb-ei-Ej) the energy is not optimised with respect to all parameters but we do know how the orbitals change because the uai are solutions of the SCF coupled-Hartree-Fock (CPHF) equations.28 The non-redundant uai (a = virtual i = occupied) are the explicit solution of these equations A ux = b" which in detail are ((er-ei) uFi+zz [4(irI sk)-(isI rk)-(ikI rs)]u&} sk =-&fr+S& ei++I z S&[4(ir(sk)-(isIrk)-(ikIrs)] (26) sk for closed-shell systems.The remaining ux may be expressed26 in terms of the solution of the CPHF equations but it is desirable to use formulae which do not explicitly evaluate uij (i and j occupied) and Uab (a and b virtual) because these depend on (ei-ej)-l and (e -eb)-l respectively. .~~ Pople et ~1have given a formula for the first derivative of EMp2,which can be rewritten in the form where Tx depends upon derivative integrals and Waiis given by Standard GAUSSIAN programs solve the CPHF equations for the ux (3N of them N = number of nuclei) and then calculate dE/dX according to eqn (27).N. C. HANDY However eqn (27) can be rewritten as dE -= U/TuX+P dX = U/rA-lb"+ T" (31) where ATZ= w. (33) If the gradient is evaluated through eqn (32) then only one set of simultaneous eqn (33) have to be solved for 2,instead of 3N; more importantly no derivative integral has to be stored because whether it contributes to T" or hx,it can immediately be multiplied by its appropriate factor and entered directly into the gradient vector. For second derivatives the advantages of this development are even more significant. The formula for the second derivative of the MP-2 energy has to take the form To evaluate this the first term which involves the solutions of uxY of the second order CPHF equations [(3w2of them] can be reduced by a similar analysis to eqn (30)-(33) to u/TuXY = FbXY (35) where bxY is the right-hand side of the second-order CPHF equations.Therefore to evaluate a2E/aX(3Y only the first-order CPHF equations have to be solved. The evaluation of MP-2 second derivatives should therefore be a relatively straightforward matter and we are currently implementing this procedure. These arguments also apply to the calculation of CI gradients and the details may be found in ref. (30). It also applies to the calculation of molecular properties. For example for correlated wavefunction calculations where the Hellmann-Feynman theorem is not obeyed it has been suggested by Diercksen and Sadlej31 that more accurate first-order properties may be obtained by calculating the energy derivative.This may be written for the dipole moment and for MP-2 calculations as =-NIP2 --XZai(al~li)-i ~@jb~~~(kI~li)+~x C a$bag(cIxlb) (36) a1 ai ijk ab ij abc which follows from eqn (32). These are to be compared with the formula derived from expectation values by for one-electron Marller-Plesset properties which are therefore different. VI. A VIEW OF QUANTUM MONTE CARL0 AND ITS COMPARISON WITH CONFIGURATION INTERACTION Recently a novel method for the calculation of electronic energies of small molecules was introduced by the authors of ref. (33)-(35) called the quantum Monte Carlo method. A few groups are now actively pursuing research in this area and it is of value to compare the current status of these calculations with the more standard ab initiocalculations.We start by briefly referring to our recent CI calculation on LiH (1Z+),36which was an attempt to answer the following question 'Using gaussian basis sets and CI how accurate an energy can be obtained for LiH?' THE LENNARD-JONES LECTURE Table 5. Energy and dipole moment of LiH at r = 3.015 bohr method E/ hartree PP ref. SCF (STO) -7.987 3 13 -6.001 5 41 SCF (numerical) -7.987 34 -6.002 5 42 SCF (numerical) -7.987 354 -6.002 5 31 NO-CI (ETO) -8.060 62 -5.965 43 PNO-CI (GTO) -8.064 705 -5.837 40 CI (ETO) -8.065 53 44 CI (GTO) -8.069 04 -5.86 36 -exact -8.070 49 -5.83 At the SCF level it is now possible to achieve an accuracy to 5 or 6 decimal places using the numerical procedure of Laaksonen et aL3' for the solution of Poisson-like equations.(It is interesting to that there are 'spurious' nodes in these numerical Hartree-Fock orbitals at long range just as in the Hartree-Fock orbitals for atoms. These are probably due to the long-range behaviour of these orbitals which is dominated by exchange effects.39) Up to 1983 the best CI calculation at Y had an error of 0.005 hartree (Meyer and Ro~mus,~~ PNO-CI calculation using a 9s5p4d/7s2pld basis set). The energies and dipole moments of LiH are given in table 5 for various calculations. We chose the basis set by optimising it for Li+ and H- and found that with 1 Is 1 lp 8d 3f and lg for Li and lOs 5p 3dand 2f for H the energy errors were as follows Li+ (0.00055) Li (0.00058) H-(0.00028) H (0.0000035).These calculations demonstrate the well known fact that more angular functions are required to represent the cusp in Li+ than in H-as far as the energy is concerned. The CI calculation involved all single and double replacements (1 3201 5 CSF) from a CASSCF with the lo 20 30 40 and In orbitals active. The resulting CI energy of -8.06904 hartree was in error by 145 x which can be attributed as 90 x basis-set error and 55 x quadruples error. The time for this calculations was 9 h on the IBM 3081D of which the 4-index transformation took 5 h. The conclusion to be drawn from this rather tedious calculation is that the basis-set problem remains with us at the CI level in a very much worse way than at the SCF level.To give a fair comparison with the quantum Monte Carlo method one of the authors of the LiH calculation (R. J. Harrison) undertook these calculations. He used the short time GFMC algorithm of Reynolds et al. with the fixed-node approximation. The method is well described in their paper. The equation which is simulated is where f = qh,vT and where at convergence q4 = dexact.The success of the method depends upon the ability to choose a good trial function yT because (a) the nodes off are the same as those of yT in this procedure and (b) the variance is crucially dependent on ry, as is evident from the term HyT/t//T in the equation and the fact that the energy E is calculated through E= fedV 1sfdV.s VT N. C. HANDY 33 Table 6. GFMC results for He using various forms for tptrial time-method trial wavefunctiona energy points steps blocks variational 1 term Jastrowb -2.883 6 --GFMC 1 term Jastrow -2.902 81 +O.OOO 83 100 50 900 variational 6 term Jastrowb -2.901 82 GFMC 6 term Jastrow -2.904 17+0.000 21 100 50 700 variational 6 term Hylleraas" -2.903 24 GFMC 6 term Hylleraas -2.903 9 & 0.000 4 200 200 12 GFMC 6 term Hylleraas -2.903 74f0.000 05 200 200 1156 a The orbital part = exp [ -1.6875(r +r,)] for the Jastrow function. From N. C. Handy (Ph.D. Thesis). Ref. (46). At the end it is hoped that the distribution of points represents the density f=fexact vT,but one has to be very careful about how long it takes to reach this ideal distribution.Years ago one of the justifications for the trans-correlated method45 was that if CO was an exact wavefunction then HCO = ECO (39) implied that (C-lHC) O = EO (40) and thus that the use of the non-hermitian operator CfHCwas not too important. However it is probably true to say that sufficiently accurate functions CO were never obtained to realise fully the potential of the method. In the GFMC method vTis represented as CO,where O is a linear combination of determinants and C is a Jastrow-like factor. For atoms Jastrow factors are or with similar expressions for molecules. C has no nodes. In table 6 some results are given for He. The results for the Hylleraas trial function show that with a large amount of time high-accuracy results can be obtained.However the results obtained with increasingly accurate Jastrow functions only marginally reduce the variance and question whether the Jastrow form is most appropriate. The results for LiH and Be are summarised in table 7. All of these used the simple Jastrow factor.41 For LiH the variational CO (SCFO) energy gives 47% of the correlation energy and the corresponding result with a 4-term MCSCF O yields 67% of the correlation energy. The best GFMC result is twice as accurate as the CI result for approximately the same cost in computing time. The disappointing aspect is that the variance is not reduced by using an MCSCF O,and is only reduced by extending the run time i.e.the number of blocks. For Be it is not surprising that the MCSCF results are far more accurate than those achieved with an SCF @ (98% instead of 88 % of the correlation energy for the best GFMC calculations). This is due to the considerable configuration mixing and thus 2 bAR THE LENNARD-JONES LECTURE Table 7. GFMC results for LiH (re = 3.015) and Be time-method energy points steps blocks LiH (i) SCF 0 -7.986 50 C Q (variational) -8.026 4 & 0.000 4 1000 20 821 GFMC (Z = 0.01) -8.069 6 f0.000 7 1000 50 99 (ii) MCSCF Q -8.013 64 --C @ (variational) -8.042 8 f0.000 5 1000 50 657 GFMC (Z = 0.01) -8.069 73 & 0.000 26 1000 50 887 (iii) CI -8.069 0 --(iv) exact -8.070 49 Be (i) SCF -14.572 4 --C 0(variational) -14.608 9&0.000 8 800 100 135 GFMC (Z = 0.01) -14.656 4+0.000 5 800 50 347 (ii) MCSCF 0 -14.628 0 -C Q (variational) -14.647 8 +O.OOO 7 800 100 124 GFMC (Z = 0.01) -14.665 1 & 0.000 4 800 100 270 (iii) exact -14.667 3 the change in the nodal surfaces.However again we notice that the variance is not reduced. The probable conclusions of these GFMC calculations and similar calculations by other workers are as follows. (a)The dominant configurations should be included in because they can affect the nature of the nodal surfaces. Their inclusion does not increase the time cost very much. (b) The Jastrow factor is obviously very important. However it is not clear how to determine the non-linear parameters. Does the shape near the electron4ectron cusp depend upon the local electron density? Is the form given by eqn (41) appropriate? (c) The principal effort must be to reduce the variance.(d) Therefore until either the Jastrow factor can be substantially improved or the variance reduced in some other way these calculations are likely to remain both enormously expensive and restricted to a very small class of molecules. VI. CALCULATIONS OF HIGH ACCURACY ON SMALL BERYLLIUM CLUSTERS There have been many calculations in recent years on small Be clusters [see e.g.ref. (47) and (48)] and the purpose here will be to demonstrate that high-accuracy results can be achieved on Be, Be and Be which are at variance with calculations of less accuracy. Be2 There is now complete agreement between theoreticians and experimentalists on the potential-energy curve for the ground state (Xxi).Indeed it is one of the successes of high-accuracy ab initio calculations that they predicted a 2.3 kcal minimum at 4.75 bohr before the experimentalists confirmed that Be existed with several vibrational levels.The definitive calculation on Be, which used a large basis set and sufficient CSF is due to Lengsfield et aZ.49Earlier calculations all suffer from a lack of one or both N. C. HANDY of these and now become largely historical. A complete list of these earlier works may be found in ref. (14). The results of these many ab initio calculations on Be suggest the following conclusions. First the SCF curve is purely repulsive and secondly SDCI calculations predict a shallow (< 1 kcal mol-l) minimum around 8.5 bohr.These statements are independent of basis set. For a 7s+ 3p+ Id basis set a shallow minimum is also predicted by SCEP CEPA-3 CEPA-1 and also CCSD and CCSDT-1. Lee and Bartlett50 have presented a clear explanation of these results showing in particular that neglect of triple and quadruple excitations leads to too deep a well at short distances (eg. CEPA-0) whilst inclusion of only doubles and quadruples produces too repulsive a curve at short distances. Thus MBPT(4) and full CI (SDTQCI) should produce satisfactory results and indeed they both predict a well of ca.0.8 kcal mol-1 around 5.0 bohr. A 7s +4p +2d full-CI calculation yields an interaction energy of 1.06 kcal/mol-l at 4.75 bohr. Our best full-CI result (8s+5p+2d+ If basis set) gave 1.86 kcal mol-1 at 4.75 bohr and is in complete agreement with the more restricted MRSDCI results of Lengsfield et al.49We have argued that further basis-set extension must lead to a final result in the region of 2.2-2.3 kcal mol-1 for the interaction energy with g functions estimated to contribute 0.15 kcal mol-l.These agree with the experimental results of Bondybey and English,51 who predict re =4.65 bohr and D in the range 2.14-2.29 kcal mol-l. Be has therefore turned out to be an excellent molecule to demonstrate the need for high-accuracy calculations which need both a large number of CSF and a large basis set. Be3 Increasing sp hybridisation means that there is an increase in the stability for the sequence Be, Be and Be,.However because of the above arguments for Be we would expect that a large calculation involving many excitations will be necessary to achieve the correct dissociation energy for Be,. At the SCF level Be (D3h)seems not to be bound although Whiteside et al.48found D,to be 1.5 kcal mol-l in a 6-31G* basis set. The SCF curve approaching from dissociation passes through a maximum before approaching the minimum. Whiteside et al. found in a MP(4) (SDQ) calculation that the binding energy was 6.0 kcal mol-1 for r =2.23 A i.e. that electron-correlation effects are significant. The best calculation we performed on Be involved a 7s+ 3p+ 2d basis set with an initial CASSCF calculation for those orbitals correlating with 2s 2px and 2py orbitals on the atoms (z out of the plane).A reference set was then chosen as all SD CSF with orbitals correlating with 2s 2px 2p and 2p,. Finally an SDCI out of this reference set generated 784498 CSF in C, symmetry. A binding energy of 19 kcal mol-l was calculated and unlike SCF there is no maximum in the potential curve. A reasonable basis-set extension would predict a binding energy of > 25 kcal mol-l. Be* The situation for Be is rather different because SCF calculations predict Be to be bound by 36 kcal mol-l. 52 Calculations for the correlation energy which include polarisation functions in the basis set show that it has a considerable effect on D,. Whiteside et al.48give 56.0 kcal mo1-I (2.126 A G).The best result of Bauschlicher et al.52is for an SDCI calculation and gives a 64 kcal mo1-1 D,at 3.92 bohr with the Davidson correction contributing 19 kcal mol-l to this result.The best calculations performed by us5 used a TZ+DP basis set for SDCI+ Davidson correction gave a binding enery of 66.1 kcal mol-l. As a confirmation of this 2-2 THE LENNARD-JONES LECTURE result CEPA-l gave 64.8 kcal mol-1 at the same geometry G,re = 3.92 bohr. A more complete basis-set calculation could easily increase D,to 75 kcal mol-l. SUMMARY OF Be, Be AND Be It is interesting to place the results for this sequence in a table cluster DJkcal mol r,/bohr symmetry 2.3 4.75 25 4.22 75 3.92 This shows the shortening of the bond as the sp polarisation effects increase. The earlier results in the literature which we have summarised above show the great care with which these calculations have to be performed before reliable results are obtained.N. C. Handy Chem. Phys. Lett. 1980,74,280. R. J. Harrison and N. C. Handy Chem. Phys. Lett. 1983,95 386. P. Saxe H. F. Schaefer and N. C. Handy Chem. Phys. Lett. 1981,79 202. R. K. Nesbet J. Chem. Phys. 1965,43 317. P. E. M. Siegbahn Chem. Phys. Lett. 1984 109 417. E. R. Davidson J. Comput. Phys. 1975 17 84. 'I J. Paldus J. Chem. Phys. 1974 61 5321. I. Shavitt Znt. J. Quantum. Chem. Symp. 1977 11 131; 1978 12 5. O B. R. Brooks and H. F. Schaefer J. Chem. Phys. 1979,70 5092. lo P. Saxe D. J. Fox H. F. Schaefer and N. C. Handy J. Chem. Phys. 1982 77 5584. l1 P. J. Knowles and N. C. Handy Chem.Phys. Lett. 1984 111 315. l2 B. 0. Roos P. E. M. Siegbahn and P. R. Taylor Chem. Phys. 1980,48 157. l3 P. J. Knowles and H-J. Werner Chem. Phys. Lett. 1985 115 259. l4 R.J. Harrison and N. C. Handy Chem. Phys. Lett. 1983,98 97. l5 C. Mnrller and M. S. Plesset Phys. Rev. 1934 46 618. S. Wilson Theoretical Chemistry (Spec. Period. Rep. The Chemical Society London 1981) vol. 4 p. 1. R. J. Bartlett H. Sekino and G. D. Purvis Chem. Phys. Lett. 1983 98 66. R. J. Bartlett and E. Brandas J. Chem. Phys. 1972 56 5467. lo J. S. Binkley and J. A. Pople Int. J. Quantum Chem. 1975 9 229. 2o W. D. Laidig G. Fitzgerald and R. J. Bartlett Chem. Phys. Lett. 1985 113 151. 21 K. Hirao and Y. Hatano Chem. Phys. Lett. 1984 111 533. 22 F. B.Brown I. Shavitt and R. Shepard Chem. Phys. Lett. 1984 105 363. 23 H-J. Werner and E. A. Reinsch J. Chem. Phys. 1982 76 3144; see also W. Meyer in Modern Theoretical Chemistry ed. H. F. Schaefer (Plenum New York 1977). 24 J. S. Binkley R. A. Whiteside R. Krishnan R. Seeger D. J. Defrees H. B. Schlegel S. Topiol L. R. Kahn and J. A. Pople Quantum Chemistry Program Exchange 1981 13,406. 25 J. S. Binkley M. J. Frisch D. J. Defrees R. Krishnan R. A. Whiteside R. Seeger H. B. Schlegel and J. A. Pople Gaussian-82 (Carnegie-Mellon University Pittsburgh). 26 J. A. Pople R. Krishnan J. B. Schlegel and J. S. Binkley Int J. Quantum Chem. 1979 S13 225. 27 P. Pulay Mol. Phys. 1969 17 197. 28 J. Gerratt and I. M. Mills J. Chem. Phys. 1968 49 1719. 20 J. D.Goddard N. C. Handy and H. F. Schaefer J. Chem. Phys. 1979 71 1525. 30 N. C. Handy and H. F. Schaefer J. Chem. Phys. 1984,81 5031. 31 G. H. F. Diercksen and A. J. Sadlej J. Chem. Phys. 1981 75 1253. 32 R. D. Amos Chem. Phys. Lett. 1980 73 602. 33 J. B. Anderson J. Chem. Phys. 1975,63 1499. 34 J. W. Moskowitz K. E. Schmidt M. A. Lee and M. H. Kalos J. Chem. Phys. 1982 77 349. 35 P. J. Reynolds D. M. Ceperley B. J. Alder and W. A. Lester J. Chem. Phys. 1982 77 5593. 36 N. C. Handy R. J. Harrison P. J. Knowles and H. F. Schaefer J. Phys. Chem. 1984 88 4852. 37 L. Laaksonen P. Pyykko and D. Sundholm Chem. Phys. Lett. 1983 % 1. N. C. HANDY 37 38 P. Pyykko personal communication. 39 N. C. Handy M. T. Marron and H. J. Silverstone Phys. Rev. 1969 180 45.40 W. Meyer and P. Rosmus J. Chem. Phys. 1975 63 2356. L'' P. E. Cade and W. M. Huo J. Chem. Phys. 1967,47 614. 42 E. A. McCullough Chem. Phys. Lett. 1974 24 55. 43 G. F. Bender and E. R. Davidson J. Phys. Chem. 1966 70 2675. 44 D. W. Bishop and L. M. Cheung J. Chem. Phys. 1983,78 1396. 45 N. C. Handy and S. F. Boys Proc. R. London Soc. A 1969 310 43. 46 E. A. Hylleraas Z. Phys 1929 54 347; 1930 60,624; and 1930 65 209. 47 P. S. Bagus H. F. Schaefer and C. W. Bauschlicher J. Chem. Phys. 1983 78 1390. 4* R. A. Whiteside R. Krishnan J. A. Pople M. Krogh-Jesperson P. von R. Schleyer and G. Wenke J. Comput. Chem. 1980 1 307. 49 B. H. Lengsfield A. D. McClean M. Yoshimine and B. Liu J. Chem. Phys. 1983 79 1891. 50 Y.S. Lee and R. J. Bartlett J. Chem. Phys. 1984 80,4371. 5* \1. E. Bondybey and J. H. English J. Chem. Phys. 1984 80 568. 52 C. W. Bauschlicher P. S. Bagus and B. N. Cox J. Chem. Phys. 1982 77 4032. 53 R. J. Harrison and N. C. Handy to be published.

ISSN:0301-5696    

DOI:10.1039/FS9841900017

出版商:RSC    

年代:1984

数据来源: RSC

摘要:

Faraday Symp. Chem. SOC., 1984 19 39-48 Silacyclobutadiene Singlet and Triplet Geometries Vibrational Frequencies and Electronic Structures E. COLVIN III* BY MICHAEL AND HENRYF. SCHAEFER Department of Chemistry University of California Berkeley California 94720 USA. Received 16th August 1984 The geometries of the lowest singlet and triplet states of silacyclobutadiene have been optimized at the double-zeta (DZ) and double-zeta (DZ + d) self-consistent-field (SCF) level of theory. Silacyclobutadiene has a planar asymmetric closed-shell ground state. The ground-state C-C and C-Si bond lengths are in good agreement with standard values indicating that this state is closely analogous to the ground state of cyclobutadiene. The calculated DZ+d SCF harmonic vibrational frequencies support this analogy.The lowest triplet state of silacyclo- butadiene is found by configuration-interaction calculations to lie only ca. 5 kcal mo1-l above the ground state much lower than the established value of 23 kcal mol-' for the singlet-triplet separation in cyclobutadiene The overall stability and singlet-triplet separation of silacyclo- butadiene are discussed in terms of bond energies and 71-system aromaticity. The last decade has seen a dramatic upsurge in experimental and theoretical interest in organosilicon chemistry. Beginning in 1976 with the synthesis and characterization of the first silicon+arbon double bond,l. this interest has increasingly focused on strained and unsaturated organosilicon c~mpounds.~ In the theoretical study of such species a natural compound to investigate is silacyclobutadiene (SiC3H4).It is a representative model species of conjugated organosilicon systems yet it is small enough to be treated by rigorous ab initio techniques. Moreover silacyclobutadiene has special theoretical relevance because its properties offer insights into the extensively studied problem of the nature of the cyclobutadiene ground state.4 Of course silacyclobutadiene is of interest in its own right. It is a new and virtually unstudied compound and hence its properties are of interest to chemists attempting to characterize it further. EXPERIMENTAL AND THEORETICAL BACKGROUND In 1983 Muetterties and Gentle announced the tentative observation of sila- cyclobutadiene by thermal-desorption mass spectroscopy during a study of the silane surface chemistry of palladi~m.~ While Muetterties' study provides the only experi- mental information on silacyclobutadiene at this time some previous theoretical results are available for this species.In 1980 Gordon reported a self-consistent-field (SCF) full-geometry optimization of silacyclobutadiene using a split-valence 3-21G basis set followed by single-point calculations with an extended 6-31G* basis set.6 He found silacyclobutadiene to be a planar asymmetrical structure stable relative to dissociation to acetylene and sila-acetylene by 56.3 kcal mol-l. Although little is known about silacyclobutadiene the analogous compound cyclobutadiene has certainly been one of the most vigorously pursued molecules of this century.' A particular emphasis of recent experimental and theoretical work on 39 STUDIES ON SILACYCLOBUTADIENE cyclobutadiene has been the elucidation of the ground-state geometry and spin- multiplicity.While it has been known for more than a decade that certain crystalline derivatives of cyclobutadiene have singlet rectangular ground conclusive results for cyclobutadiene itself have proved more elusive. Initial infrared studies of cyclobutadiene detained in an argon matrix led to the conclusion that it possessed a square ground state,12-14 seemingly indicative of a triplet state. However numerous failures to observe an e.s.r. signal argued strongly against a triplet ground state.8 This discrepancy was finally resolved in favour of the rectangular singlet ground state by Masamune using i.r.analysis of cyclobutadiene photosynthesized in an argon matrix.l5 Concurrent with this experimental work has been a tremendous theoretical effort by many groups. For theoreticians cyclobutadiene is of interest not only because of the controversial nature of its ground state but also because it is a model system for the study of bond strain steric hinderance n-conjugation and aromaticity. Theoretical studies of cyclobutadiene have almost unanimously predicted a rec- tangular singlet ground ~tate.l~-~~ While this result disagreed with the early i.r. studies mentioned above the e.s.r. data (or lack thereof) cast some doubt on the possibility of a triplet ground state.Later semiempirical and more sophisticated two-configuration SCF studies of this problem confirmed the ground state to be a rectangular singlet,18-21v 23 which was consistent with the e.s.r. data but contradicted the available i.r. results. This disagreement instigated a number of fairly extensive configuration- interaction (CI) investigations in an attempt to resolve what for several years appeared to be a major discrepancy between theory and experiment. Although the subsequent i.r. results proved the early experimental conclusions wrong these CI studies were valuable inasmuch as they gave a solid theoretical picture of the lowest states of cyclobutadiene. For example the a+n CI calculation of Jafri and Newton using a 6-3lG* basis predicts a rectangular singlet ground state 23.0 kcal mol-1 more stable than the square triplet state.21 The square singlet transition state was found to lie 12.0 kcal mol-1 above the ground state.The goal of the current study was to establish comparable or perhaps even more reliable theoretical results for the lowest-energy states of silacyclobutadiene. THEORETICAL METHODS The stationary points for the lowest singlet and triplet states of silacyclobutadiene were located and characterized at the self-consistent-field (SCF) level of theory. These theoretical studies were carried out using two basis sets. The first basis set used was the relatively standard double-zeta (DZ) basis of Huzinaga and D~nning~~-~~ with the following contraction scheme Si(l1s 7p/6s 4p) C(9s 5p/4s 2p) H(4s/2s).The second basis set (DZ +d) was the Huzinaga-Dunning DZ basis described above augmented with 6 d-like functions [x2,y2,z2,xy xz yz times exp (-ar2)]centred on silicon (a = 0.6) and carbon (a = 0.75). The starting point for the geometry optimization was the 3-21G SCF structure of Gordon.6 The optimization was facilitated by the use of analytic SCF gradient tech.niques with an optimization criteria that the energy gradients be < hartree bohr-l.? The harmonic vibrational frequencies of the DZ and DZ+d singlet structures were obtained via finite differences of analytic SCF gradients. Similarly the i.r. intensities were evaluated from dipole derivatives determined by finite differences of dipole moments calculated at the displaced geometries.-f 1 hartree = 4.359814 x lo-'* J; 1 bohr = 5.291771 x lo-" m. M. E. COLVIN AND H. F. SCHAEFER I11 H H DZ DZ+d Fig. 1. Predicted one-configuration SCF equilibrium geometries for the silacyclobutadiene ground state. All bond lengths in A. Two-configuration SCF (TCSCF) single-point calculations were carried out on the ground singlet state of silacyclobutadiene at the optimized one-configuration geometry. The two configurations corresponded to the orbital occupancies (2~”~3a”~) and (2~”~4a”~). In addition pi-space and single- and double-(SD) excitation CI calculations were carried out at the SCF equilibrium geometries. The pi-space CI involved single and double CI with all but the valence a’’ orbitals held frozen. Specifically the orbitals held frozen were Si( Is 2s 2px 2py 2pz 3s 3px 3py) C( Is 2s 2px 2p,) and H( 1s) and the corresponding virtual orbitals.This resulted in 9 1 and 435 configurations respectively for the DZ and DZ+d singlet state and in 123 and 627 configurations for the DZ and DZ +d triplet states. In the SDCI calculation the core orbitals on silicon (Is 2s 2ps 2pv 2pz) and carbon (1s) were held frozen. This larger CI involved 25771 and 79603 configurations for the DZ and DZ+d singlet state respectively and 29355 and 91 779 configurations for the DZ and DZ +d triplet state. EQUILIBRIUM STRUCTURES The predicted SCF equilibrium geometries for the lowest singlet and triplet states of silacyclobutadiene are given in fig. 1 and 2. The large differences in silicon-carbon bond lengths in going from the DZ to DZ+d basis sets (up to 0.1 A for the singlet state) illustrate the necessity of including d-functions on the heavy atoms.The predicted DZ +d bond lengths for singlet silacyclobutadiene suggest that this structure should be interpreted in terms of conventional carbon-carbon and silicon- carbon single and double bonds. The two carbon-carbon bond lengths in the singlet are predicted to be 1.539 and 1.346 A in close agreement with the standard values of ca. 1.54 A for a carbon-carbon single bond and ca. 1.34A for a carbon-carbon double bond.29 The longer of the predicted silicon-carbon bonds is 1.866 A in comparison with a value of 1.867 A determined from the microwave spectrum of methyl~ilane.~~ The shorter of the silicon-carbon bonds in silacyclobutadiene is predicted to have a length of 1.688 A in good agreement with the result of 1.698 A reported for the silaethylene silicon-carbon bond by an earlier theoretical study using a similar basis set and level of theory.31 Similarly the reported theoretical cyclobutadiene structures show strong agreement of the carbon-carbon bond lengths STUDIES ON SILACYCLOBUTADIENE H .H DZ DZ+d Fig.2. Predicted SCF equilibrium geometries for the lowest triplet state of silacyclobutadiene. All bond lengths in A. with the standard Thus it is reasonable to conclude on structural grounds that the singlet ground state of silacyclobutadiene is closely analogous to the ground state of cyclobutadiene. The SCF equilibrium structures of the lowest triplet state of silacyclobutadiene (fig.2) do not allow as clear an interpretation. Nevertheless a few obvious points are interesting to note. First the effect of the added d-functions is much less pronounced in the triplet state (maximum DZ to DZ + d change 0.051 A) than in the singlet state (maximum change 0.096 A). Secondly the triplet state does not have a diagonal axis of symmetry so that structurally it is not exactly analogous to the square lowest triplet state of cyclobutadiene. Thirdly the shortest carbon-carbon and carbon-silicon bonds are not on opposite sides of the ring as they are in the singlet indicating a dramatic difference in the electron distributions between the two states. The differences in the lengths for the comparable bonds in the singlet and triplet states offer some insight into the nature of the 3A’ state.The most dramatic changes in going from the lA’ to the 3A’ state are in the silicon-carbon ‘double bond’ which expands by 0.158 to1.846 A and in the carbon-carbon ‘single bond’ which shrinks by 0.102 A to 1.437 A. In contrast the carbon-carbon ‘double bond’ expands by only 0.036 A to 1.382 A and the silicon-carbon ‘single bond’ shrinks by only 0.037 A to 1.829 A. The overall result of these changes is that the silacyclobutadiene 3A’ state silicon-carbon bonds have lengths comparable to a Si-C single bond (1.846 and 1.829 A as compared with 1.867 A in methyl~ilane),~~ while carbon-carbon bond lengths are closer to that of a carbon-carbon double bond (1.437 and 1.382 A as compared with 1.339 in ethylene).32 As noted earlier the silicon substitution rules out a truly square structure for triplet silacyclobutadiene.Nevertheless the triplet structure is much more ‘square-like ’ in geometry than is singlet silacyclobutadiene. VIBRATIONAL FREQUENCIES The DZ + d SCF harmonic vibrational frequencies (cm-l) and the corresponding infrared intensities D (A2a.m.u.)-l for the lA‘ ground state are given in table 1 along with descriptive assignments. A comparison of the predicted frequencies of the normal modes corresponding to the various heavy-atom bond stretches with the experimental stretching frequencies of similar bonds yields more information about the nature of these bonds. While making these comparisons it should be kept in mind that several studies of DZ and DZ + d SCF harmonic vibrational frequencies have found that the M.E. COLVIN AND H. F. SCHAEFER I11 Table 1. Vibrational frequencies for the singlet ground state of silacyclobutadiene predicted at the DZ +d SCF level of theory symmetry frequency/cm-l intensity/DZ(A~a.m.u.1-1 description U’ 3485 0.0 C-H stretch U’ 3462 0.1 C-H stretch U’ 3353 0.7 C-H stretch U’ 2424 3.9 Si-H stretch U’ 1621 2.2 C=C stretch U’ 1360 0.2 C-H in-plane rock U’ 1228 1.4 C-H in-plane rock U’ 1150 1.4 C-PI in-plane rock U’ 1090 0.2 Si=C stretch a’’ 1042 0.1 C-H out-of-plane wag U’ 888 2.0 C-C stretch U’ 861 0.9 C-H Si-H rock U’ 762 0.3 C-H out-of-plane wag U“ 694 2.7 Si-H in-plane rock a’’ 541 0.1 C-H out-of-plane wag U’ 522 0.2 Si-C stretch U” 503 2.3 ring fold U” 306 1.o Si-H out-of-plane wag Table 2.Comparison of selected DZ +d SCF harmonic vibrational frequencies with analogous experimental results frequencies/cm-] ~ ~~~~ mode predicted experimental C-C stretch 888 ca. 1000 (cy~lobutadiene)’~ 995 (ethane)37 C=C stretch 1621 1523 (cy~lobutadiene)’~ Si-C stretch 523 cu. 700 (methyl~ilane)~~ Si=C stretch 1090 1001 1003 (1,l -dimethyl~ilaethylene)~~? 40 986 (m~nomethylsilaethylene)~~ predicted frequencies are systematically ca. 10% higher than the experimentally measured This error arises because correlation and anharmonic effects are not included at such levels of theory. The predicted frequencies of selected normal modes and the bond stretches they correspond to are given in table 2 along with the comparable experimental results for various molecules.For the modes corresponding to the stretching of the carbon+arbon single and double bonds and the siliconsarbon double bond the qualitative agreement is good (including the anticipated 10% error). The predicted frequency for the silacyclobutadiene silicon5arbon single bond stretch is 523 cm-l ca. 100 cm-l short of what it should be to agree well with the experimental result of 7OOcm-l for methyl~ilane.~~ This discrepancy is probably a result of some mixing of the predicted mode with other low-frequency ring modes so that it does not correspond to a simple STUDIES ON SILACYCLOBUTADIENE Table 3.Total energies for singlet and triplet silacyclobutadiene (in hartree) singlet triplet level of theory DZ DZ+d DZ DZ+d single configuration two-configuration SCF -404.607 25 -404.626 70 -404.699 96 - -404.625 71 - -404.707 60 - pi-space CI single- and double-excitation CI -404.682 13 -404.913 76 -404.755 40 -405.1 19 87 -404.666 60 -404.913 83 -404.9 13 83 Table 4. Singlet-triplet energy separations in silacyclobutadiene ~ level of theory relative energies/kcal mol-l singlet triplet DZ DZ+d single configuration single configuration -1 1.58 -4.79 two-configuration single-configuration 9.58 -pi-space CI pi-space CI 9.75 -CISD CISD 0.0 1.7 Davidson Davidson 3.3 3.5 Si-C stretch. The overall good agreement of these predicted frequencies with the experimental frequencies for the free bonds corroborates the result derived from structural considerations that the carbon+arbon and silicon-carbon single and double bonds in the lA’ ground state of silacyclobutadiene are not greatly perturbed by ring strain or conjugation effects.SINGLET-TRIPLET ENERGY SEPARATION The total and relative energies for the lowest singlet and triplet states of sila- cyclobutadiene are given in tables 3 and 4. For comparison some predicted singlet- triplet separations for cyclobutadiene are given in table 5. At the single-configuration SCF level of theory the silacyclobutadiene 3A’ state is predicted to be 11.6 and 4.8 kcal mol-1 more stable than the lA’ for the DZ and DZ +d basis sets respectively.This result closely matches the cyclobutadiene single- determinant 6-31G* singlet-triplet separation of 5.8 kcal mol-1 predicted by Hehre and Pop1e.l’ The fact that single-determinant SCF theory predicts the triplet state to be lower in energy is not surprising. A one-determinant RHF wavefunction includes in a certain sense some electron correlation for triplet states due to the ‘Fermi hole’ between the triplet-coupled electron~.~~ Moreover the low-lying 4a”orbital (occupied in the triplet state) is neglected in the one-configuration treatment of the closed-shell lA’ state. To describe better the singlet state a two-configuration wavefunction was employed with the second configuration corresponding to the 3a”2-P 4a”2 excitation. The weights for these two configurations are 0.94 (2aN23d2) and -0.33 (2a”24a”2).The relatively large coefficient of the second configuration demonstrates the importance of this configuration in describing the singlet state. Numerous studies have demonstrated the veracity of singlet-triplet energy separations M. E. COLVIN AND H. F. SCHAEFER 111 Table 5. Published singlet-triplet energy separations in cyclobutadiene singlet-triplet theoretical method separation /kcal mol-l ref. 6-31 G* single configuration SCF -5.8 17 STO-3G pi-space CI 6-31G* U+Z CI 22.4 23.O 22 21 predicted by comparing a two-configuration closed-shell singlet with the corresponding single-configuration triplet ~tate.~~-~~ Such a DZ calculation for sila- cyclobutadiene finds the TCSCF singlet 9.6 kcal mol-1 more stable than the triplet.In light of the dramatic effect of the second configuration on the singlet-state energy we decided to investigate further the singlet-triplet energy separation using configuration interaction. Studies of cyclobutadiene have found that the results of larger CI calculations could be accurately anticipated with a much smaller pi-space CI calculation considering only excitations among valence a’’ orbitals.22,24 Single-point pi-space CI calculations on the lowest singlet and triplet states of silacyclobutadiene (at the SCF equilibrium geometries) are in good agreement with the results using the two-configuration singlet/one-configuration triplet scheme (see table 4). A large SDCI calculation of the singlet-triplet energy separation somewhat decreases the pi-space CI results and leads to a final predicted singlet-triplet splitting of ca.5 kcal mol-l. For comparison the most reliable predictions of the singlet-triplet separation in cyclobutadiene are 21 kcal (Kollmar and Staemmlerls) 22 kcal (Borden et aZ.22)and 23 kcal (Jafri and Newton21). That the singlet-triplet energy separation for silacyclobutadiene is ca. 17 kcal mol-1 less than that for cyclobutadiene can be understood in terms of bond-energy considerations. As discussed earlier the rectangular singlet ground states of cyclo- butadiene and silacyclobutadiene contain essentially conventional single and double covalent bonds. Thus it is reasonable to consider these species in terms of the energies of the individual bonds.Several earlier studies have shown that silicon+arbon and carbon*arbon single-bond energies are very similar (ca.88 kcal mol-l for Si-C47 and ca. 85 kcalmol-l for C-C49. However this similarity does not hold for the com- parable n-bonds. On the basis of bond strength and kinetic data Walsh recommends a silicon-carbon n-bond energy of 39 5 kcal m~l-’,~~ nearly 20 kcal mol-l less than the well established carbon-carbon n-bond energy of ca. 57 kcal11101-~.~~ This means that the disruption of the cyclobutadiene n-system in going from the singlet to the triplet state comes at greater energetic cost than in silacyclobutadiene and hence the greater singlet-triplet energy gap. The same result is seen in the relative singlet-triplet energy separations of ethylene and silaethylene.In the 1978 paper of Hood and Schaefer and the more recent CEPA study of Kohler and Lischka the silaethylene singlet-triplet splitting was predicted to be 38.5 and 35.2 kcal mol-l respectively in comparison with a much higher value of 62.8 kcal mol-1 for ethylene.49 A question of interest in the study of cyclobutadiene is whether the square closed-shell singlet state lies at a sufficiently low energy to act as a transition state between the two possible identical rectangular forms of cyclobutadiene (i.e. the two possible orientations of the double bonds). As mentioned earlier the square singlet tran- sition state is found to be more stable than the corresponding triplet state by STUDIES ON SILACYCLOBUTADIENE ca.11 kcalmol-l. In an attempt to determine whether an analogous state of silacyclobutadiene lies below the first triplet state we carried out a single-point DZ TCSCF energy calculation for the lA’ state at the optimized ,A‘ geometry. The ‘square’ singlet was found to be 0.8 kcal mol-1 more stable than the corresponding triplet state. Note however that full optimization of the singlet transition state may either increase or decrease this square singlet-triplet gap. SILACYCLOBUTADIENE STABILITY Although silacyclobutadiene has been shown to be a relative minimum on the potential surface (there are no imaginary vibrational frequencies) the overall thermo- dynamic stability of silacyclobutadiene is still open to question. One of the goals of Gordon’s study of silacyclobutadiene was to determine if this species was stable with respect to dissociation to acetylene and sila-acetylene.Gordon6 carried out this calculation at the SCF level of theory using 6-31G* basis sets at the 3-21G equilibrium geometries. He found silacyclobutadiene to be 56.3 kcal mol-1 more stable than the separated products. We repeated this calculation with a DZ basis set and found a somewhat lower result of 44.8 kcal mol-I. While these results are in qualitative agreement with the results of Trinquier50 and others51 that when possible the preferred forms of unsaturated organosilicon compounds are rings the question of the silacyclobutadiene stability requires closer scrutiny. A recent study of one of these dissociation products sila-acetylene has shown that it is a stationary point but not a minimum on the potential surface.52 Instead linear sila-acetylene is predicted to have a degenerate imaginary mode corre- sponding to bending to yield the shallow trans bent minimum.At the Davidson- corrected SDCI level of theory silylidene is reported to be 43 kcal mot1 more stable than sila-acetylene with a barrier to interconversion of 49 kcal mol-l. Taking this into account silacyclobutadiene is predicted to be ca. 5 kcal mol-l less stable than the dissociation products acetylene and silylidene. However since the initial concerted reaction of silacyclobutadiene to form acetylene and sila-acetylene is symmetry f~rbidden,~, the barrier to unimolecular dissociation should be considerable.SILACYCLOBUTADIENE AROMATICITY Since silacyclobutadiene is essentially an annulene the possibility of aromatic stabilization or destabilization by n-electron conjugation should be considered. The thermochemical concept of aromaticity is derived from the observation that certain annulenes with (4n +2) n-electrons (e.g.benzene) possess special stability because of their conjugated n-~ystems.~ Associated with this concept of aromatic stabilization is the idea that 4n n-electron annulenes are destabilized by cyclic conjugation (so-called antiaromaticity). However note that there is no specific evidence that disruption of the cyclic conjugation would stabilize such annulenes. In fact experimental studies of one 4n annulene 1,5-bisdihydroannulene (n = 3) indicate that this system prefers a nearly planar conformation even at the expense of increased angular It is difficult to judge the exact role of aromatic effects on silacyclobutadiene.While the ground state is planar and stable with respect to out-of-phase distortions the geometry would seem to preclude significant n-conjugation (see structure section). In his study of silacyclobutadiene Gordon predicted what he called the ‘anti- aromaticity ’ using the bond-separation formula55 SiC,H +3CH +SiH,-+CH SiH +CH CH +CH SiH +CH CH,. (1) M. E. COLVIN AND H. F. SCHAEFER 111 47 Gordon calculates AE for the above reaction to be -53.5 kcal mol-1 and states that this 'is an indication of strong antiaromaticity in silacyclobutadiene. ' By comparison cyclobutadiene is found to have a corresponding AE of -67.9." Using eqn (1) we calculate the DZ SCF silacyclobutadiene antiaromaticity to be 49.1 kcal mol-l.These results indicate strong antiaromatic destabilization but they are not entirely unambiguous. In particular an obvious shortcoming of this bond-separation scheme is its inability to distinguish destabilizing conjugation effects from angle 57 and 1,3 interaction^^^ known to be important in four-membered ring systems. CONCLUDING REMARKS The ground state of silacyclobutadiene has been determined to be planar closed-shell singlet analogous to the rectangular ground state of cyclobutadiene. The bond lengths and vibrational frequencies of the singlet indicate little n-conjugation in the silacyclobutadiene ring.The lowest triplet state lies ca. 5 kcal mol-1 above the ground state. This research was supported by the U.S.National Science Foundation Chemistry Division grant no. CHE-8218785. We are also grateful to Energy Conversion Devices Inc. for an unrestricted grant in support of this research. Helpful discussions with Drs Jozef Bicerano and John de Neufville (ECD) and Mark Gordon (North Dakota) were much appreciated as was theoretical help from Dr Michel Dupuis. 0. L. Chapman C. C. Chang J. Kolc M. E. Jung J. A. Lowe T. J. Barton and M. L. Tumey J. Am. Chem. Soc. 1976,98 7844. M. R. Chedekel M. Skoglund R. L. Kreeger and M. Shechter J. Am. Chem. SOC.,1976 98 7846. For a review of experimental work see L. E. Gusel'nikov N. S. Nametkon and V.Vdovin Acc. Chem. Res. 1975 8 18. For recent reviews see G. Maier Angew. Chem. 1974 86 491; Angew. Chem. Int. Ed. Engl. 1974 13,425; T. Balley and S. Masamune Tetrahedron Rep. 1980,36,343; M. P. Cava and M. J. Mitchell Cyclobutadiene and Related Compounds (Academic Press New York 1967). T. M. Gentle and E. L. Muetterties J. Am. Chem. SOC.,1983 105 304. M. S. Gordon J. Chem. Soc. Chem. Commun. 1980 1131. ' Beginning with A. Kekule Liebigs Ann. Chem. 1872 162 77. L. T. J. Delbaere M. N. G. James Nakamura and S. Masamune J. Am. Chem. SOC.,1975,97 1973. R. S. Brown and S. Masamune Can. J. Chem. 1975 53 972. M. Irngartner and M. Rodewald Angew. Chem. 1974,86 783; Angew. Chem. Int. Ed. Engl. 1974 13 740. l1 G. Lauer C. Miller K. W. Schulte A.Schweig and A. Krebs Angew. Chem. 1974,86 597; Angew. Chem. Int. Ed. Engl. 1973 13 544. l2 A. Krantz C. Y. Lin and M. D. Newton J. Am. Chem. SOC.,1973,95 2744. l3 0. L. Chapman C. L. McIntosh and J. Pacansky J. Am. Chem. Soc. 1973 95 614. l4 0.L. Chapman D. De La Cruz R. Roth and J. Pacansky J. Am. Chem. Soc. 1973,95 1337. lBS. Masamune F. A. Souto-Bachiller T. Machiguchi and T. E. Bertie J. Am. Chem. Soc. 1978 100 4889. l6 M. J. S. Dewar and A. Komornicki J. Am. Chem. Soc. 1977 99 6174. l7 W. J. Hehre and J. A. Pople J. Am. Chem. Soc. 1975 97 6941. H. Kollmar and U. Staemmler J. Am. Chem. Soc. 1977 99 5383. l9 M. J. S. Dewar M. C. Kohn and N. Trinajstic J. Am. Chem. SOC.,1971 93 3437. 2o M. J. S. Dewar and H. Kollmar J. Am. Chem. Soc. 1975,97 2933.21 J. A. Jafri and M. D. Newton J. Am. Chem. Soc. 1978 100 5012. 22 W. T. Borden E. R. Davidson and D. Hart J. Am. Chem. SOC.,1978 100 388. 23 N. L. Allinger and J. C. Tai Theor. Chim. Acta 1968 12 29. 24 M. Nakayama N. Nishihira and Y. I'Maya Bull. Chem. SOC.Jpn 1976 49 1502. 25 S. Huzinaga J. Chem. Phys. 1965 42 1293. 26 T. H. Dunning J. Chem. Phys. 1970 53 2823. 27 S. Huzinaga and Y. Sakai J. Chem. Phys. 1968 50 1371. 48 STUDIES ON SILACYCLOBUTADIENE 28 T. H. Dunning and P. J. Hay Mod. Theor. Chem. 1977 3 1. 29 S. H. Pine J. B. Henrickson D. J. Cram and G. S. Hammond Organic Chemistry (McGraw-Hill New York 1980) p. 87. 30 R. W. Kilb and L. Pierce J. Chem. Phys. 1957 27 108. 31 H. F. Schaefer Ace. Chem. Res. 1982 15 283.32 J. L. Duncan I. J. Wright and D. Van Lerberghe J. Mol. Spectrosc. 1977 42 463. 33 Y. Yamaguchi and H. F. Schaefer J. Chem. Phys. 1980 73 2310. 34 M. Colvin G. P. Raine and H. F. Schaefer J. Chem. Phys. 1983 79 1551. 35 T. J. Lee and H. F. Schaefer J. Chem. Phys. 1984,80 2977. 36 G. P. Raine and H. F. Schaefer J. Chem. Phys. in press. 37 T. Shimanouchi Tables of Molecular Vibrational Frequencies Natl Stand. Ref. Data Ser. (Natl Bur. Stand. U.S.A. 1972) vol. 1. 38 T. Shimanouchi J. Phys. Chem. Rev. Data 1977 6 993. 39 L. E. Gusel’nikov V. V. Volkova V. G. Avakyan and N. S. Nametkin J. Organomet. Chem. 1980 201 137. 0.M. Nefedov A. K. Maltsev V. N. Khabashesku and V. A. Korolev J. Organomet. Chem. 1980 201 123. 41 T. J. Drahnak J. Michl and R.West J. Am. Chem. SOC. 1981,703 1845. 42 A. Szabo and N. S. Ostlund Modern Quantum Chemistry Introduction to Advanced Electronic Structure Theory (MacMillan New York 1982). 43 R. R. Lucchese and H. F. Schaefer J. Chem. Phys. 1978 68,769. 44 C. F. Bender J. H. Meadows H. F. Schaefer Faraday Discuss. Chem. SOC. 1977 62 59. 45 P. J. Hay W. J. Hunt and W. A. Goddard Chem. Phys. Lett. 1972 13 30. 46 C. F. Bender H. F. Schaefer D. R. Franceshetti and L. C. Allen J. Am. Chem. SOC. 1972,94,6888. 47 R. Walsh Acc. Chem. Res. 1981 14 246. 48 K. W. Egger and A. T. Cocks Helv. Chim. Acta 1973 56 1516. (a)D. M. Hood and H. F. Schaefer J. Chem. Phys.. 1978,68,2985; (b)H. J. Kohler and H. Lischka J. Am. Chem. SOC. 1982 104 5884. 50 G. Trinquier and J. P. Malrieu J.Am. Chem. SOC. 1981 103 6313. 51 R. S. Grev and H. F. Schaefer J. Chem. Phys. 1984 80 3552. 52 M. R. Hoffman Y Yoshioka and H. F. Schaefer J. Am. Chem. SOC. 1983,105 1084. 53 R. B. Woodward and R. Hoffmann The Conservation of Orbital Symmetry (Verlag Chemie Weinheim 1970). 54 R. Gygax J. Wirz J. T. Sprague and N. L. Allinger Helv. Chim. Acta 1977 60,2522. 55 W. J. Hehre R. Ditchfield L. Radom and J. A. Pople J. Am. Chem. SOC. 1970 92 9796. 56 P. v. R. Schleyer J. E. Williams and K. R. Blanchard J. Am. Chem. SOC. 1970 92 2377. 57 I. J. Miller Tetrahedron 1969 25 1349. 58 M. L. Bauld J. Cessac and N. L. Holloway J. Am. Chem. SOC. 1977 99 8140. (PAPER 19/2)

ISSN:0301-5696    

DOI:10.1039/FS9841900039

出版商:RSC    

年代:1984

数据来源: RSC

摘要:

Faraday Symp. Chem. SOC.,1984 19,49-61 Potential-energy Surfaces for Chemical Reactions Dimerization of CH and SiH, the SN2Reaction in Gas-phase Clusters and CH Activation in Transition-metal Complexes BY UIJI ~TSUHISAOHTA,NOBUAKI SHIGERU MOROKUMA,* KOGA OBARA~ AND ERNEST R. DAVIDSON~ Institute for Molecular Science Myodaiji Okazaki 444,Japan Received 3rd September 1984 We present the results of three applications of the molecular-orbital method to problems of potential-energy surfaces that control chemical reactions. In the first application the dimerization of CH and SiH in both their singlet and triplet states is investigated in connection with the path of least motion as against the path of non-least motion. Two ground-state (,B,) methylenes in the path of non-least motion give ground-state ethylene with no barrier.The ground-state (lA,) silylenes give a ground-state disilene with a barrier in the path of least motion and no barrier in the path of non-least motion. In the second problem potential-energy surfaces are calculated for an S,2 reaction (H,O),OH-+CH,Cl -+ HOCH +C1-+nH,O where the reactants are complexed with one or two water molecules. When the hydroxide ion is solvated by two water molecules the reaction takes place through the first step of reactant complex formation followed by inversion of the methyl group. The transition state for methyl inversion has an energy comparable to that of the reactants. The migration of water molecules from the hydroxide side to the chloride side is not involved in the rate-determining process.The last problem is concerned with the activation of an inert CH bond in transition-metal complexes. In six-coordinate Ti do complexes the optimized geometry shows theoretical evidence that the distortion of the ethyl or methyl ligand represented by a short M. -H distance a small M...C-C (or H) angle and a long CH bond is of electronic origin. Electronegative axial ligands are essential for the existence of an agostic interaction which is stabilized by CH a+Ti dzy charge transfer. A similarly distorted ethyl group has been found in the calculation of a three-coordinate Pd complex. The low-energy transition state for p-elimination lies along a smooth extension of the ethyl distortion. 1. INTRODUCTION One of the goals of theoretical chemistry has been a full understanding of the mechanisms rates and dynamics of complicated chemical reactions.Potential-energy surfaces for ground and excited states control reactions and their theoretical studies have been a focal point since the early days of molecular-orbital theories. However in the past such studies have been confined to very qualitative model calculations or to very small (e.g.triatomic) systems. In the last ten years we have seen much progress in this area. The energy-gradient or more generally energy-derivative method is the largest contributor to these activities.l The energy-derivative method has provided quantum chemists with the power to explore efficiently complicated highly multi- 1' Present address Department of Chemistry Faculty of Science Kyoto University Kyoto Japan.$ Present address Department of Chemistry Indiana University Bloomington Indiana 47405 U.S.A. 49 POTENTIAL-ENERGY SURFACES dimensional potential-energy surfaces determine their characteristic features such as equilibrium geometry saddle-point (transition-state) geometry and the point of avoided crossing and characterize such properties of these surfaces as the intrinsic reaction coordinate and seams of crossing. Optimization of the equilibrium structure for molecules having 10 or more atoms is becoming routine and the determination of the transition state for systems of this size is feasible with a reasonable amount of computer time. Since its methodological developments will no doubt be discussed in other papers in this symposium we will restrict ourselves to applications of the method to calculations of potential-energy surfaces.In this paper we present the results of our studies of potential-energy surfaces with regard to three different topics. In the next section we discuss the reactions of both the triplet and singlet states of methylene and silylene i.e. CH +CH -+ CH,CH SiH +SiH -+ SiH,SiH CH +SiH +CH,SiH by comparing the paths of least motion and non-least motion. In the third section we present potential-energy surfaces for an SN2reaction in a hydrated cluster (H,O),OH-+CH,Cl +HOCH +C1-+nH,O (n = 0 1,2). The fourth section deals with the activation of an inert CH bond in transition-metal complexes in particular the equilibrium structure of some Ti complexes and the transition state for @-elimination in a Pd complex.We end with a brief conclusion. 2. LEAST-MOTION versus NON-LEAST-MOTION PATHS FOR THE DIMERIZATION OF CH AND SiH The dimerization of singlet methylenes to form ground-state ethylene had been considered as a textbook example of a reaction path of non-least motion. In the least-motion path (D,,)two o orbitals of CH units (oA,oB) become a ogand a nu orbital in ethylene and ah ok -+ oiniis symmetry-forbidden. In the path of non-least motion proposed by Hoffmann Gleiter and Mallory (hereafter referred to as H.G.M.)2 the reaction path starts with C symmetry mixing o and n orbitals and making the process symmetry-allowed and takes the higher symmetry of D, in a later stage of the reaction.This argument however is based on an extended Hiickel method i.e. a single-determinant wavefunction which is insufficient to describe essential electronic configurations. Moreover the ground state of CH is a triplet ,B1,to which the above argument does not apply. Recent ab initio calculations have shown that two ground-state triplet methylenes can dimerize via the path of least motion without a barrier to give ground-state eth~lene.~ On the other hand two singlet methylenes dimerize via the path of least motion to give a Rydberg excited state of ethylene.* Ohta et aL5 have recently studied the dimerization of singlet methylenes and triplet methylenes via a path of non-least motion. They have also studied the dimerization of triplet and singlet silylene (SiH,) both via the path of least motion and a non-least-motion path.The ground state of silylene is a singlet IAl and the lowest triplet ,B1 is the first excited state the opposite of the case of methylene. The coupling reaction of CH and SiH to give silaethylene CH,=SiH, via the least-motion path and a non-least-motion path has also been investigated. The basis set used is of double-zeta +polarization quality and for ethylene a set of Rydberg-type sp functions has been added. Calculations are mainly K. MOROKUMA K. OHTA N. KOGA S. OBARA AND E. R. DAVIDSON -77.70 -1.5 2.0 2.5 3.0 3.5 4.0 R(C-C)/A Fig. 1. Potential-energy curves for dimerization of CH via the path of non-least motion as functions of the CC distance.carried out with a CAS (complete active space) MCSCF wavefunction including all (20) configurations for 4 active electrons in 4 active orbitals (a,o*,n,n*). Potential curves for the excited states of CH +CH are calculated with multi-reference (MR) CI including all the single and double excitations (6332 spin-adopted configurations) from the four active orbitals to all virtual orbitals. 2. I. NON-LEAST-MOTION DIMERIZATION OF CH +CH We have used the path of non-least motion determined by H.G.M. and augmented it with a few more points at a short CC distance in D,,symmetry. At the far end of this path where the overall symmetry is C, one CH has its C, axis nearly parallel to the line of approach whereas the other CH has its C, axis nearly perpendicular to the line of approach.At a CC separation of 2.45 A the bending angles of the two CH units become equal with overall C, symmetry and a trans conformation. At 2.00 the bending angles become zero and the entire system is planar (D,,). The structure of the CH fragments is fixed at R(CH) = 1.10 A and LHCH = 120" throughout the path. In fig. 1 the potential-energy curves along the path of non-least motion for the ground and some excited states are shown. The ground-state singlet along the curve starts with two triplet (3B1) methylenes and goes without a barrier to the ground state of ethylene. The situation is essentially the same with the least-motion dimerization but differs from the H.G.M. results where two singlet (lA,) methylenes form the ground-state ethylene.The dimerization of two singlet (lA,) methylenes in fig. 1 follows the first excited state which is a valence excited state at long distance passes over a barrier caused by avoided crossing and becomes a Rydberg excited state of ethylene. This situation is again similar to the case of the least-motion path. The third and fourth states represent the dimerization of CH,(lA,) +CH,(lB,). They are nearly degenerate to ca. 2.3 A below which the lower state forms excited ethylene without a barrier. 2.2. SiH +SiH +SiH,SiH Fig. 2 shows four potential-energy curves for overall singlet states for the dimerization of SiH,. Curves (a)-(c) are for the least-motion paths and curve (d)is POTENTIAL-ENERGY SURFACES -580.00-0 c 2 -580.05 -2 1 ril -580.0-I I I I 2.0 3.0 4.0 5.0 R(Si-Si)/A Fig. 2. Potential-energy curves for dimerization of SiH as functions of the SiSi distance. for a path of non-least motion. In (a)the SiH distance and the HSiH angle are fixed at the calculated values for the lA ground state of SiH, i.e. 1.497A and 93.9'. This represents the dimerization of singlet silylenes in the path of least motion. Fig. 2 and an analysis of the wavefunction indicate that in the first half of the dimerization reaction the potential curve is repulsive as the SiSi distance decreases and the total wavefunction (singlet) consists mainly as expected of a product of a2 singlet wavefunctions of the reactants. At ca. 3.1 A the triplet (an)x triplet (an) configuration takes over which leads to the ground-state disilene.The barrier is caused by avoided crossing between two major configurations and is a typical example of symmetry-forbidden reactions. Curves (6) and (c) both assume the SiH distance and HSiH angle fixed at the calculated values for 381 SiH, i.e. 1.460 A and 118.0'. In our calculation the singlet is lower in energy than the triplet even at this geometry. Therefore the lower curve (b) as in (a) consists mainly of a singlet x singlet configuration and becomes triplet x triplet inside the barrier owing to avoided crossing at ca. 4.0 A. The barrier is earlier in (b)than in (a),because the assumed fragment geometries in (b)are more favourable to the triplet x triplet configuration than the singlet x singlet configuration.The upper curve (c) although determined only as the second root of the CI equation in the MCSCF procedure represents the least-motion path for dimerization of 3B silylenes. The triplet x triplet wavefunction at a long distance goes through an avoided crossing with (b)and is adiabatically correlated to an excited state of disilene. The path of non-least motion for dimerization of singlet SiH was determined by the Hartree-Fock-Roothaan (H.F.R.) optimization of geometrical parameters as functions of the SiSi distance. At an early stage the SiH units have a singlet-like structure; one has its C,,axis nearly parallel to the line of approach while the others are perpendicular. In the final stage there are two SiH units with triplet-like structure trans bending angle becoming small.The MCSCF potential-energy curve along this path is shown in fig. 2(d). There is no barrier along this path starting from singlet silylenes. In conclusion the ground-state singlet SiH dimerizes to form the ground state of K. MOROKUMA K. OHTA N. KOGA S. OBARA AND E. R. DAVIDSON -328.9 t I -328.95 I--329.051 / I I I I 2.0 3.0 4.0 R(C-Si)/A Fig. 3. Potential-energy curves for CH +SiH -+ CH,SiH as functions of the SIC distance. disilene without a barrier on the path of non-least motion and with a substantial barrier on the least-motion path. Therefore the H.G.M. conclusion is applicable to the dimerization of SiH,. The excited-state triplet SiH is led adiabatically to an excited state of disilene but actually is likely to form ground-state disilene with no barrier through a non-adiabatic transition in the least-motion path.If the triplet silylene is lower in energy than the singlet silylene at the optimum geometry of the triplet a non-adiabatic transition will be unnecessary and the path of least motion will give the ground-state product without a barrier. 2.3. CH +SiH -+ CH,SiH At infinite separation for this mixed system our calculations give the total energy of the singlet pairs as slightly lower in energy than the triplet pairs. Therefore as shown in fig. 3 the lowest least-motion path is curve (a) where the fragment geometries are fixed at the optimum values for CH,(lA,) and SiH,(lA,). Curve (a) is repulsive as all the least-motion singlet x singlet curves are and goes over a large barrier at ca.3.2 A to reach the ground-state product. In curves (b)and (c) the fragment geometries are frozen at the values in the equilibrium geometry of silaethylene CH,SiH, which are not far from those in the triplet fragments. The lowest curve (b) essentially describes the least-motion reaction for triplet pairs which reaches the ground-state silaethylene with no barriers. Curve (d)represents the potential-energy curve on the path of non-least motion which was determined by H.F.R. geometry optimization as a function of the CSi distance. The singlet pairs form the ground-state product without a barrier. Therefore the overall qualitative situation for this mixed system is similar to the case of SiH +SiH, although the detailed features of the potential curves are different.Note that in the early stage of the path of non-least motion the preferred geometry POTENTIAL-ENERGY SURFACES has the SiH plane nearly perpendicular to the line of approach and the CH plane nearly parallel to it as though SiH were acting as an electron acceptor and CH as an electron donor. An analysis reveals that in the early stage of the reaction electrostatic interactions are dominant and it is most favourable to have the large CH dipole aligned parallel to the line of approach. 3. POTENTIAL-ENERGY SURFACES FOR THE SN2 REACTION IN GAS-PHASE CLUSTERS Chemical reactions in gas-phase clusters are attracting considerable attention. Not only are cluster reactions interesting in themselves but their mechanisms rates and dynamics should provide information on solvent effects and may fill the gap between reactions in the gas phase and in so1ution.6y7 The S,2 reaction in solvated clusters is one of the few reactions in which rate constants in the cluster have been determined for various numbers of solvent molecules as well as in the gas phase and in solution.For instance the rate constant for the reaction (H,O),OH-+ CH,Cl+ HOCH + C1-+nH,O is fastest in the gas phase; it is slower in clusters for n = 1 2 and 3 by ca. .6 500 and 5000 times respectively while in solution it is lo1 times s10wer.~ Previously we have carried out calculations of potential-energy surfaces for the following symmetric S,2 reactions:* (H,O),Cl-+ CH,Cl + ClCH + Cl-(H,O), (n = 0 1,2).Our findings can be summarized as follows. (i) The most favourable reaction path for n = 1 is reactants -+ reactant complex + transition state for CH inversion -+ transfer of H,O from the left (the newly formed CH,Cl side) to the right (the newly formed C1- side) -+ product complex -+ products. Since the system is symmetric the process by which H,O transfer takes place before CH inversion is equally favourable. The path of simultaneous CH inversion and H,O transfer is both energetically and entropically unfavourable. (ii) For n = 2 the most favourable path is reactants -+ re-actant complex + transfer of one H,O molecule from the left (Cl-) to the right (CH,Cl) -+CH inversion -+transfer of the other H,O molecule from left (newly formed CH,Cl) to the right (newly formed C1-).Having one water molecule on each chlorine is the best way to stabilize the intrinsically symmetric transition state for CH inversion of reaction (2). (iii) The transfer of H,O from one side to the other takes place with little or no barrier via an intermediate having a bent Cl.-*C.*-Cl configuration. However the situation wth the experimentally studied reaction (1) could be substantially different. While reaction (2) is thermoneutral reaction (1) is highly exothermic (AEPexptl = -47.5 kcal mol-l). Because of high exothermicity the path following the transition state for CH inversion is expected to be downhill and the system loaded with much released energy is expected to shake off solvent molecules.The transition state for reaction (2) is symmetric and just at the midpoint between the reactants and products whereas that for reaction (1) is expected to be ‘early’. Because both the electronic structure and geometry of the transition states are expected to be different the number of solvent molecules on each site (OH or Cl) could be different between the two near their respectively favourable transition states. K. MOROKUMA K. OHTA N. KOGA S. OBARA AND E. R. DAVIDSON 0 -10 OH -+ C H3 CI OH-+CH3CI-W -20 I -0 E -30 W-OH'+CH-JCI I 6 0 .r 4 W-OH-+CH3CI-W -40 '-reaction coordinate Fig. 4. Potential-energy profiles for W,OH-+CH,ClW + [W,OH.**CH;**ClW,]- W = H,O. Ohta and Morokumag have recently studied potential-energy surfaces for the following five SN2reactions (W denotes H20) OH-+ CH,C1 -+ [OH.**CH,..*Cl]--+ products (3) OH- +CH,ClW -+ [OH*.-CH;-CIW]-+ products (4) WOH-+ CH,Cl + [WOH-..CH;.*Cl]-+ products (5) WOH-+ CH,ClW -+ [WOH...CH,.**ClW]-+ products (6) W20H-+ CH,Cl -+ [W,OH***CH;**Cl]-+products.(7) According to previous calculations8? lo the absolute value of the barrier height in S,2 reactions is very sensitive to the basis set used in particular to polarization and diffuse functions but depends little on electron correlation. Therefore we used the 6-3 lG* basis set augmented with C1 and 0 anionicp functions for CH,Cl and OH- and the 6-31G basis set for solvent water molecules. The geometries of the reactants reactant complex and transition state for CH inversion/transfer were optimized and their energies were calculated with the H.F.R.energy gradient under the restriction of C symmetry. The profiles of the potential-energy surfaces are shown in fig. 4. Since the interaction energy with H20 is much larger for OH- than for CH3C1 the profiles for reactions (4) and (6) are similar to those of reactions (3) and (5),respectively. At the transition state for CH inversion some net charge develops on the chloride. Therefore the water molecule on the chloride stabilizes the system and lowers the barrier from the reactant complex substantially for reactions (4) (calculated barrier height 1.O kcal mol-1 and (6) (2.3 kcal mol-l) as compared with reactions (3) (2.8 kcal mol-l) and (5) (4.7 kcal mol-l) respectively.In the dihydrated systems (6) and (7) the transition state for CH inversion for reaction (7) has a substantially lower energy than that for POTENTIAL-ENERGY SURFACES H2.95 H995 108.9 0 108.6 0 10.99 Il.01 6 172.3H 5 172.1~ / / H 1.68:’ H 4,95 Hq.95 108.8 0 109.1 0 10.99 8 172.6H 7 172.1 H/loo U 1.68; ,109.5 H ,’1-61 103 7 H CI h Fig. 5. Geometriesof reactant complexes and transition states. Compound numbers correspond to those in fig. 4. reaction (6). Therefore the reaction of W20H-should proceed first through the transition state for CH inversion without transferring water molecules. Migration of water molecules from the newly formed CH,OH to C1-will take place after this transition state during an exothermic energy release so that H20migration will not be involved in the rate-determining step.This situation is in contrast to the case of CI-+CH,Cl where H,O migration is an important part of the rate process. The barrier height for reaction (7) is large (9.5 kcal mol-l) and the energy of the transition state is as high as that of the reactants. This is consistent with the experimental finding that the rate of reaction decreases abruptly at n = 2 suggesting that the overall barrier relative to the isolated reactants must have become positive. The geometries of reactant complexes and transition states are shown in fig. 5. The location of the transition state relative to the reactant complex and the product complex can be related to the change in exothermicity upon hydration.Because of space limitation further discussion is omitted here. K. MOROKUMA K. OHTA N. KOGA S. OBARA AND E. R. DAVIDSON / 12.571) D' H Fig. 6. Optimized geometries of Ti(C,H,)(PH,),(Cl),(H) and Ti(CH,)(PH,),(Cl),. 4. CH ACTIVATION IN TRANSITION-METAL COMPLEXES Activation of inert CH bonds by transition-metal complexes is a topic of current interest in organometallic chemistry and homogeneous catalysis. X-ray and neutron diffraction studies have identified several complexes in which the hydrogen atom in a CH bond is located unusually close to the metal centre,ll indicating a direct interaction between the metal and a CH bond. The transition metals involved are of a wide variety including Ti Mn Fe Cu Mo Ru Rh Pd and Ta.In all cases an electron count shows that the central metal is electron deficient and the inclination to satisfy the 18-electron rule has been considered to be a necessary condition for interaction. This kind of hydrogen atom called an agostic hydrogen has been taken as evidence for the incipient activation of an inert CH bond by a transition metal. However experimental evidence is limited to the structures of stable complexes and there is little direct evidence of agostic hydrogen having a high reactivity in intramolecular hydrogen-migration reactions. 4.1. DISTORTED ETHYL AND METHYL GROUPS IN SIX-COORDINATE Ti do COMPLEXES Recently Koga et aZ.13 found the first theoretical evidence of an agostic hydrogen in an ab initio calculation.The optimized structure of the six-coordinate Ti do complex Ti(C,H,)(PH,),(Cl),(H) 1 shown in fig. 6 was obtained using the H.F.R. method with a double-zeta-quality basis set for the valence electrons of Ti and C,H and a minimal set for the core electrons and other ligands. Note that the distance between Ti and one p hydrogen of the ethyl group is very small 2.23 A indicating a direct interaction between them. The TiCC angle is 89" substantially less than the standard tetrahedral angle expected in an sp3-hybridized carbp atom and the CHB bond distance involved in the interaction 1.11 A is 0.03 A longer than the other CH bonds. These structural features are in agreement with the X-ray diffraction results for Ti(C,H,)(drnpe)(Cl), 2 [dmpe = dimethylphosphinoethane P(CH,),CH,CH,P-(CH,),] which are shown in fig.6 in parentheses.12 The present calculation indicates that these unusual structural features are of electronic origin not caused by crystal-packing forces. An analysis of the wavefunction reveals that the complex has low-lying vacant molecular orbitals consisting of Ti dxy,to which electron delocalization takes place from the CHP bonding orbital. Additional calculations indicate that the distortion of the ethyl group is sensitive to POTENTIAL-ENERGY SURFACES Table 1. Dependence of geometrical parameters of Ti(CH,)(PH,),(X),Y on ligands X and Y and PTiP angles X Y LPTiP/" LTiCH1/' P/" R(TiC)/A R(TiH1)/A H H H H 91.60pa 75.0as 108.3 107.1 110.6 110.5 2.135 2.122 2.685 2.657 H C1 89.40~ 106.2 110.6 2.135 2.653 C1c1c1 Hc1c1 87.90~ 88.60~ 75.0as 102.6 100.2 99.6 109.8 109.2 108.9 2.094 2.102 2.085 2.566 2.533 2.510 a op optimized; as assumed.\ /p Fig. 7. Contour map of the LUMO of 2. Solid and dotted lines denote positive and negative values respectively. the choice of ligand. In both Ti(C,H,)(PH,),(H),(Cl) and Ti(C,H,)(PH,),(H),(H) the ethyl group is found to be undistorted with a large M*.-Hdistance of 3.01 A a normal CCTi angle of 114" and normal CH distances. The axial chlorides are essential to the distortion. A similar distortion has been found by Obara et al.14for the corresponding methyl compound. The optimized structure for Ti(CH,)(PH,),(Cl),(Cl) 3 is also shown in fig.6 in which the PTiP angle was fixed at 75" to simulate the situation for the experimentally studied Ti(CH,)(dmpe)(Cl), 4. The TiCHl angle is 99.6" substantially smaller than the standard tetrahedral angle or the other TiCH angle (1 13.1"); i.e. the CH group is distorted. The angle D between the pseudo-three-fold axis of CH and each CH bond is 109* indicating that the tetrahedral methyl group as a whole is twisted away from the standard three-fold axis. An earlier structure for 4 determined by X-ray diffraction giving a TiCHl angle of 70" turned out to be unreliable.' The latest neutron diffraction results,12 shown in parentheses in fig. 6 give LTiCHl = 93.7" and R(Ti**-H)= 2.45 A in reasonable agreement with our theoretical prediction. K.MOROKUMA K. OHTA N. KOGA S. OBARA AND E. R. DAVIDSON 1626 H H H Fig. 8. Geometries of reactant 5 (R) product 6 (P) and the transition state (T) for the p-elimination reaction. The extent of distortion is sensitive to both axial and cis ligands X and Y as well as the PTiP angle as shown in table 1. Starting with a nearly undistorted CH for X =Y = H the TiCHl angle is decreased by the replacement of X or Y by C1 and a decrease in the PTiP angle. The most effective is the axial ligand C1 as in the case of the above-mentioned ethyl analogue 1. The donative interaction from the CH bond to an unoccupied Ti dzyorbital appears to be responsible for this distortion as in the case of the ethyl distortion. Fig. 7 shows the LUMO of 3 at the optimized geometry of fig.6. This mainly consists of a Ti dxy orbital with some mixing of s and p orbitals and extends to the direction x =y to take in the out-of-phase CH bonding orbital. This implies that in the occupied molecular-orbital space the CH bonding orbital will have a small portion of this Ti vacant orbital mixed in-phase contributing to direct M*-.H bonding. 4.2. AGOSTIC HYDROGEN AND FACILE /I-ELIMINATION IN THREE-COORDINATE Pd COMPLEXES Agostic hydrogens in the above-mentioned Ti complexes are not reactive since the complexes already have six ligands and the agostic hydrogens therefore cannot be transferred to metal centres. Reactive intermediates without sufficient ligands would not be stable enough to make themselves available to experimental structural analysis.Theoretical calculations capable of predicting the geometries of stable complexes should be able to give structural and energetic information as to the existence and behaviour of agostic hydrogens in unstable reactive intermediates. Koga et al.15 have recently found in an ab initio calculation a three-coordinate intermediate Pd(C,H,)(PH,)(H) 5 to have an agostic hydrogen. The path of /I-elimination to give an ethylene complex 6 CHZ-CHz H 2CyCH2 I \ H-Pd-PHj I H Pd-PHj I I H H 5 6 POTENTIAL-ENERGY SURFACES Table 2. Barrier for p-elimination and insertion reactions in kcal mol-1 method p-elimination insertion H.F.R. 11.0 8.0 M P2 2.8 5.8 has been found to be on a smooth continuation of the agostic H**-M interaction and to have a low barrier.The basis sets used are valence double-zeta and core minimal for Pd 3-21G for atoms in the ethyl group and STO-2G for the PH group. The geometry optimization was carried out at the H.F.R. level. The optimized structure of the intermediate 5 is shown in fig. 8. The PdCC angle is 88O the Pd*..H distance is short (2.13 A) and the CH distance directly involved in the interaction 1.13 A is 0.05 A longer than other CH distances in the same ethyl group. All these features as before point to a direct Pd*.*H interaction. A comparison between this three-coordinate Pd intermediate and the six-coordinate Ti complex discussed in section 4.1 indicates that the ethyl group in this complex is more deformed than in the Ti complex despite the smaller electron deficiency in the former (electron count 14) than in the latter (electron count 12).This suggests that the presence oi’ three-coordination in a Pd d8complex which prefers four-coordination provides an empty ‘site’ convenient for interaction and is a factor favouring the agostic interaction. The optimized structure for the /?-elimination product 6,given in fig. 8 is that of a typical ethylene complex. The geometry of the transition state for /?-elimination is shown also in fig. 8. The Pd...H distance 1.65 A is closer to that of the product (1.59 A) than that of the reactant (2.13 A). The C-C distance 1.40 A is closer to the 1.34 A of the product than the 1.53 A of the reactant. These results indicate that /?-elimination has a ‘late’ transition state.Note also that the Pd-C bond is not as stretched as in the product which suggests that this transition state is ‘tight’ as well as being ‘late’. Table 2 shows the barrier heights for /?-elimination and its reverse the insertion reaction calculated with the H.F.R. and MP2 methods at H.F.R.-optimized geometries. The reaction in either direction has a rather low barrier suggesting the reversibility of the /?-elimination/insertion.16 Experimentally an equilibrium between ethyl complexes and ethylene complexes has also been observed in some cases.17 A low activation enthalpy for the insertion reaction has been observed in an Rh complex.lS Our calculated results for a model Pd complex suggest that such a facile /?-eliminationlinsertion may be taking place via an intermediate having an agostic hydrogen.It has been shown experimentally that when the ethyl group has electronegative substituents the /?-elimination reaction is suppressed. The geometry optimization for Pd(CH,CHF,)(PH,)(H) Pd(CH,CF,)(PH,)(H) and the transition state connecting the two gives an undistorted difluoroethyl group in the reactant intermediate and a high barrier. The electron-withdrawing fluorine atom on the /3 carbon makes the electron-donating ability of the CH bond too small for a favourable agostic interaction. 5. CONCLUDING REMARKS Thanks to the development of the energy-derivative method and other advances in theory and coding the way in which quantum chemists explore multidimensional potential hypersurfaces has changed dramatically.With these new developments K. MOROKUMA K. OHTA N. KOGA S. OBARA AND E. R. DAVIDSON quantum chemists can carry out chemistry. We no longer have to be bound to unrealistically simplified model compounds or model reactions. Rather we can deal with realistic if not real reactants and reactions with reasonable reliability and confidence. In the coming years molecular-orbital calculations may help an experimentalist to conceive a non-existing molecule and examine its structure stability reactivity and other properties theoretically before he/she actually goes to the laboratory to synthesize it. One may also be able to design or control a chemical reaction leading to a specific product by calculating the transition states and barriers of reactions and by evaluating alternatives.The design of homogeneous catalysts by changing the metals and ligands on the computer and finding a most promising route might be feasible. We are grateful to Dr Kazuo Kitaura who is a coauthor of one of the papers on which the presented work is based. Numerical calculations were carried out at the Computer Centre of I.M.S. E. R. D. was a visiting professor at I.M.S. when the work presented here was carried out. P. Pulay in Applications of Electronic Structure Theory ed. H. F. Schaefer (Plenum Press New York 1977) p. 153 and references therein; A. Komornicki K. Ishida K. Morokuma R. Ditchfield and M. Conrad Chem. Phys. Lett. 1977 45 595; S. Kato and K. Morokuma Chem. Phys. Lett. 1979 65,19;J.D. Goddard N. C. Handy and H. F. Schaefer,J. Chem. Phys. 1979,71,1525;B. R. Brooks W. L. Laidig P. Saxe J. D. Goddard Y. Yamaguchi H. F. Schaefer J. Chem. Phys. 1980,12,4652; R. Krishnan H. B. Schlegel and J. A. Pople J. Chem. Phys. 1980,72,4654; J. A. Pople R. Krishnan and H. B. Schlegel Znt. J. Quantum Chem. Symp. Ser. 1979 13 225; Y. Osamura Y. Yamaguchi P. Saxe M. A. Vincent J. F. Gaw and H. F. Schaefer Chem. Phys. 1982 72 131; Y. Osamura Y. Yamaguchi and H. F. Schaefer J. Chem. Phys. 1982 77 383. R. Hoffmann R. Gleiter and F. B. Mallory J. Am. Chem. SOC. 1970 92 1460. H. Basch J. Chem. Phys. 1971 55 1700; L. M. Cheung K. R. Sundberg and K. Ruedenberg Int. J. Quantum Chem. 1979 16 1103; K. Ruedenberg M. W. Schmidt M. M. Gilbert and S. T. Elbert Chem.Phys. 1982 71,41. D. Feller and E. R. Davidson J. Phys. Chem. 1983 87 2721. K. Ohta E. R. Davidson and K. Morokuma to be submitted. K. Tanaka G. I. Mackay J. D. Payzant and D. K. Bohme J. Am. Chem. Soc. 1981 103 978; W. N. Olmstead and J. I. Brauman J. Am. Chem. SOC. 1977 99 4219; M. J. Pallerite and J. I. Brauman J. Am. Chem. SOC. 1980 102 5993. D. K. Bohme and G. I. Mackay J. Am. Chem. SOC. 1981,103,978;D. K. Bohme and A. B. Raksit J. Am. Chem. SOC. 1984,106 3447; M. Henchman J. F. Paulson and P. M. Hierl J. Am. Chem. Soc. 1983 105 5509. K. Morokuma J. Am. Chem. SOC. 1982 104 3732; K. Morokuma S. Kato K. Kitaura S. Obara K. Ohta and M. Hanamura in New Horizons of Quantum Chemistry ed. P-0. Lowdin and B. Pullman (Reidel Dordrecht 1983) p.221. K. Ohta and K. Morokuma to be submitted. lo F. Keil and R. Ahlrichs J. Am. Chem. Soc. 1976,98,4787;S. Wolfe D. J. Mitchell and H. B. Schlegel J. Am. Chem. SOC. 1981 103 7692; 7694. l1 M. Brookhart and M. L. H. Green J. Organomet. Chem. 1983 250 395 and references therein. Z. Dawoodi M. L. H. Green V. S. Mtetwa and K. Prout J. Chem. SOC. Chem. Commun. 1982,802 1410; M. L. H. Green and J. M. Williams personal communication. l3 N. Koga S. Obara and K. Morokuma J. Am. Chem. SOC. 1984 106,4625. l4 S. Obara N. Koga and K. Morokuma J. Organomet. Chem. 1984 270 C33. l5 N. Koga S. Obara K. Kituara and K. Morokuma to be submitted. l6 D. L. Thorn and R. Hoffman J. Am. Chem. SOC. 1978 100 2079. l7 J. K. Kochi Organometallic Mechanisms and Catalysis (Academic Press New York 1978).l8 D. C. Roe J. Am. Chem. SOC. 1983 105 7770. H. C. Clark J. Organomet. Chem. 1980,200,63; H. C. Clark C. R. Jablonski and C. S. Wong Znorg. Chem. 1975,14,1332; H. C. Clark C. R. Jablonski J. Halpern A. Mantorani and T. A. Weil Znorg. Chem. 1971,13,1541;G. Yagupsky C. K. Brown and G. Wilkinson J. Chem. Soc. Chem. Commun. 1969 1244.

ISSN:0301-5696    

DOI:10.1039/FS9841900049

出版商:RSC    

年代:1984

数据来源: RSC

摘要:

Faraday Symp. Chem. SOC.,1984 19 63-77 Calculation of Molecular Spectra Electronic Transitions Vibrational Patterns and Radiative Lifetimes of Spin-allowed and Spin-forbidden Transitions BY SIGRID D. F’EYERIMHOFF Lehrstuhl fur Theoretische Chemie Universitat Bonn WegelerstraDe 12 D-5300 Bonn 1 West Germany Received 21st August 1984 The use of large-scale configuration-interaction calculations for a purely ab initio description of the details of molecular spectra is discussed. It is shown that the theoretical tool is applicable to a large range of problems and that it produces quantitative data such as transition energies to within 0.2 eV or better independent of the wavelength region fine-structure effects due to spin-orbit coupling vibronic features in large-amplitude motion and radiative lifetimes of excited states on the to lo3 s timescale.In addition it gives insight into the electronic structure and the origin of the various processes; this is important for an understanding and the prediction of features which occur when for example first-row atoms in molecules are replaced by their second-row analogues or for radiation processes in competition with intersystem crossing. Ab initio calculations have become very efficient and quantitatively reliable over the years and they can now be employed in molecular spectroscopy as an alternative to experimental techniques. Even though they do not predict energies to the same level of accuracy as high-resolution spectroscopy they possess various important advantages.They are completely general and hence are able to treat any wavelength region and any molecule positive or negative ion (as long as the compounds are small i.e. possess not more than six atoms other than hydrogens) in any electronic state regardless of the physical or chemical stability of the system. Furthermore the calculations can give the various electronic states over the entire geometrical range of structural parameters i.e. the entire potential-energy surface which is essential for an understanding of the photochemistry. Various properties of excited states can also be extracted relatively simply from the calculations even though not too much emphasis has been placed on this area so far presumably because there is very little experimental competition in this case.In recent years considerable effort has been made to account theoretically for fine-structure effects in the spectra (primarily those arising from spin-orbit coupling) or to treat relativistic effects in heavy atoms such as XeF or Xe on a purely ab initio basis. In what follows various examples of our work will be given carried out by groups at Bonn and Wuppertal the latter group under the direction of R. J. Buenker. METHOD All studies were carried out using the Bonn-Wuppertal MRD-CI package plus supplementary programs developed over recent years within our laboratories. The various features of our MRD-CI method which is a configuration-driven technique have been described in the literat~re.l-~ The configuration space thereby consists of 63 CALCULATION OF MOLECULAR SPECTRA all single and double excitations with respect to a set of reference configurations (multi-reference set) generally between 10 and 40.Since large A0 basis sets are required for spectral calculations this MRD-CI space expands very fast to lo6 configurations or more a number which is too large to be handled routinely in an economical way on standard general-purpose computers. Hence the MRD-CI space is partitioned into strongly interacting and weakly interacting subspaces based on the energy contribution of each configuration (relative to the energy of the reference set alone) evaluated numerically ; the subspace of strongly interacting configurations (routinely 10000-16000 which contribute more than a predetermined ‘selection threshold’ T = hartree) is treated directly to obtain the lowest-energy solutions while the energy contribution of the more weakly interacting species is accounted for in a perturbative manner based on their numerical values for energy lowering evaluated at the earlier partitioning stage.This partitioning technique is effective because the combined effect of the weakly interacting functions although large in number is quite small. A direct comparison *between the full CI energy of -25.227 6274 hartree obtained for BH in a medium-sized A0 basis,4 and the energies resulting from truncated subspaces show that errors are 0.005208 hartree 0.000285 hartree 0.000 151 hartree and 0.000053 hartree if only 586 10389 13880 or 17049 configurations respectively are selected5 from the total space of 132686 configurations.Hence the extra computational expenditure which is necessary to treat the total space is in no way related to the gain in accuracy one obtains thereby in particular since (a) only energy differences are important for spectroscopy problems and (b)errors introduced by other omissions (such as restrictions in the A0 basis) may be larger. Finally the full CI is estimated by analogy with the formula for the contribution of triple and quadruple excitations given by Davidson6 as E(ful1 CI est.) = E(MRD-CI) +(1 -Zcf) [E(MRD-CI) -E(ref.)] or generally E(ful1 CI est.) = E(MRD-CI) +AAE whereby the sum is taken over all reference configurations. For the factor (1 -ZcX) various other formulae are also found in the literature.la It is important in this connection that the correction AE to the MRD-CI energy is small otherwise the perturbative nature is not justified.In actual calculations it is found that transition energies are essentially the same regardless of whether the MRD-CI or full CI estimates are employed provided the contribution of the reference sets as measured by Xcf is approximately the same (and > 90%) in both states involved,’ i.e.both states are described in a balanced manner. If its magnitude differs in the two states by a few percent the energies from the estimated full CI are generally the more reliable. The MO basis in the MRD-CI calculations consists generally of SCF MO of the same or of different states depending on the application or computer time available; use of only one set is more economical.It has been found that the parent SCF MO are only required if a large core (doubly occupied MO) is maintained simply because in those cases relaxation of the core cannot be accomplished by the CI. In some cases it is convenient to employ natural orbitals in order to achieve a more compact representation of the MRD-CI expansion (fewer reference configurations). These are generated by diagonalizing the first-order density matrix obtained in the first MRD-CI step. Experience shows that calculated transition energies are generally within 0.1 eV or less at the full CI level if different sets of MO or NO are employed which is a good indication that the calculation level is close to the full CI in which the type of orbital transformation within a given A0 basis is immaterial for the results.Finally the A0 basis for excited states is larger than in ground-state calculations S. D. PEYERIMHOFF 65 of the same accuracy. In order to represent the excited states it is often necessary to optimize additional functions for those excited atomic states in which dissociation is possible. For calculations on excited states of MgNa+ for example 3s,4s 3p 4p and 3d functions for magnesium and sodium had to be generated; ArH calculations required the correct description of hydrogen 2s 2p 3s 3p and 3d and argon 4s 5s 4p and 3d states by a number of contracted Gaussians. This procedure is often time- consuming and it would be advantageous if not only basis sets for ground states and polarization functions were available in the literature but also those for the actual excited atomic species.Calculation of Rydberg states requires additional long-range functions but since the nature of a Rydberg state (hydrogen-like) is simple generally one or at most two functions suffice for the description of a Rydberg member. Negative ions require at least semi-diffuse p functions but if the electron affinity must be obtained to an accuracy of 0.2 eV or better semi-diffuse d functions are also needed according to our experience with for example C Si and F. In summary the A0 basis requires careful expansion over the double-zeta plus polarization routine basis generally employed for ground-state calculations.A reduction in size will invariably lead to errors in the overall treatment. In this connection the advantage of the partitioning technique of the MRD-CI space again becomes apparent. It allows the use of large A0 basis sets and hence maintains good overall accuracy but is nevertheless computationally quite feasable; the alternative route to decrease the number of A0 functions or reference configurations in order to save computer time would lead to reduced accuracy in the overall treatment. POTENTIAL-ENERGY SURFACES FOR GROUND AND EXCITED STATES The calculation of potential-energy surfaces is in principle straightforward in the MRD-CI procedure. For simple predominantly vertical transitions the energy AEe between the two electronic states at a given geometry can be related to the absorption (or emission) maximum and deviations from measured values are usually 0.2 eV or less using standard A0 basis sets.For unknown transitions such theoretical values are good guidelines for further experimental work or for the clarification of assignments. Numerous examples are given in the literature. lay 9-11 Of more interest is an investigation of the crossing of potential curves. Experimentally such a situation can be deduced only from perturbations in the spectrum but determination of the character of the possible perturber or the form of its energy surface is involved and tedious. On the other hand the calculations are a powerful tool for obtaining this information. If the interacting states are of different spin or spatial symmetry the calculations (separate for each state) are straightforward.Typical examples in which the calculations were instrumental in explaining the observations are Fi and Cli. In Cli no vibrational quantum numbers could be assigned to the upper state in the A 211,-X 2JIg transition because the vibrational levels of 211upresented 'a chaotic distribution'.12 Calculati~nsl~~ l4 have shown that.. .n4,ni 0 states 2A and "C; lie in the same energy region and cross A ",(. . .xi@) very close to its minimum and must thus be considered to be the prime perturber responsible for the complex pattern. Furthermore while in Cli the crossing point is found to the right (larger bond lengths) of the 211u minimum which allows in principle for large coupling of vibrational states (all three states possess a similar minimum energy) it is placed to the left (repulsive branch) in Ft (the minima of 2X and 2Au lie much lower than that of A TI,)and hence the perturbation together with the smaller spin-orbit coupling in Fi relative to Cli is considerably less in Ft ; this in turn explains why a fairly regular vibrational pattern could be observed in the equivalent 211,-2JIg transitions.The 3 FAR CALCULATION OF MOLECULAR SPECTRA perturbing states have also been analysed in a number of simple hydrideslO7 l5 such as in PH PH+ SiH and SH in which predissociation is an important factor as a result of the interaction. In the predissociation of +Z; in 0; the interacting states 4Zi and 411g are known but ab initio calculations16 were able to determine details of the radiationless transition by studying the spin-orbit interaction responsible for the actual magnitude of interaction.A more complicated interaction theoretically and even less likely to be detected in measurements is the crossing of states of the same symmetry. From a computational point of view care must be taken that the set of reference configurations is balanced with respect to both states. This often requires a smaller MRD-CI partitioning (selection) threshold i.e. a larger MRD-CI subspace which has to be diagonalized. Furthermore the completely different character of interacting states sometimes causes problems for the choice of the MO basis. Experience shows that it is better to use MO which are optimal for neither one nor the other state than the parent MO of one state; alternatively averaged natural orbitals for the two (or more) interacting states are an adequate choice.Finally it is not always clear whether the Born-Oppenheimer approximation on which the CI calculations are based is still valid in this crossing area. However the ratio between the energy AEof closest approach and the vibrational frequency we in the higher potential well can be used as an indication in this regard. In various causes the calculations have shown that what was thought to belong to two different progressions is in reality due to one transition into a double-minimum well of one of the states. A prime examplelo is the B 2Z+ and C 2C+state of SiH which in reality is a double-minimum state 502n2in character at small SiH separations and 50~60at large distances.17 The mixing between Rydberg and valence states is essential for the interpretation of absorption and emission spectra of molecules such as HF,18 HC119-22and Cl,.239 24 In each case the repulsive inner-branch of an intravalence shell (ionic) state with a minimum at large internuclear separation cuts through the entire Rydberg-state manifold possessing minima at small bond lengths and causes various avoided crossings with members of the same symmetry.As a result the otherwise regular pattern of a Rydberg family is heavily perturbed as experimentally observed for HCl and HF without any e~planation~~ (‘lack of theoretical guidance has hampered the analysis of the spectrum’) and various new minima appear with Rydberg-like character at the outer-branch and valence-shell character at the inner- branch of the potential well.The intense progression in C1 absorption between 78000 and 8 1000 cm-l can be explained by such a newly built Rydberg valence well 2 lZ;(a; n34pn-0 $ ou),and the fluorescence between 50000 and 75000 cm-l by the double-minimum in the 1 state with the complementary coefficients of the same configurations. A further example comes from the rare-gas diatomics. In this case the various Rydberg states originating from ou,n, nu and o MO interact since they show different behaviour with internuclear separation excitation out of the strongly antibonding ouinto non-bonding Rydberg orbitals leads to potential minima; Rydberg states originating from bonding o and nulead to a family of repulsive states whereby for Ne, for which an extensive study has been performed,26 the 0,-+ R potential curves are more repulsive than those of Rydberg states originating from nu.The situation with n is more complex. In Ne it is weakly antibonding and the corresponding Rydberg states depopulating ng may be weakly bound. The various interactions are essential for an understanding of the behaviour of Ne in its excited states and can to some extent be generalized to other noble-gas dimers. Even though potential curves have so far only been discussed in connection with electronic spectra knowledge of them is very important for various other related phenomena.Two of them can be seen in fig. 1 which shows potential-energy curves S. D. PEYERIMHOFF CI -1 155.70 \ \\ -j 3.0 4.0 5.0 6.0 7.0 8.0 RcPc,(atomic units) Fig. 1. Calculated MRD-CI potential-energy curves for C1 or C1- removal from CF,Cl,. for CF,Cl as a function of C-Cl removal. First the lowest excited-state curves,' are all repulsive and hence it is clear that photodecomposition of CF,Cl favours release of C1 atoms. This is of course one of the problems of propellant gases like CF,Cl since Cl atoms can undergo chain reactions in the stratosphere and then con- tribute to the destruction of the ozone layer. The second phenomenon apparent from fig. 1 is dissociative electron attachment at almost thermal energies under release of negatively charged chlorine ions.The negative-ion curve crosses that of the neutral system close to the latter's minimum and thus there is the possibility of intersystem crossing from the neutral to the negative-ion system whereby the latter is almost repulsive CF,Cl-Cl-. Indeed in experiments for dissociative electron 29 a C1- peak is found at 0.7 eV with an appearance potential as low as 0.1 eV; this is in perfect agreement with the ab initio work which shows the crossing to be not too far above the CF,Cl minimum. It is of interest that this process must be considered in competition with release of C1 if the balance of the 0 reactions is considered. The CF,Cl energy at small C-Cl separation which lies above that of the neutral system has been calculated by a modified MRD-CI procedure developed for short-lived to electron resonance^.^^ This method has also been applied succe~sfully~~ the 211g(7ctng)and 211n,(7t3,3s2)resonance states of N and is generally applicable to short- lived negative-ion states as is the MRD-CI method; it uses a simple method to stabilize the discrete component of a resonance state by increasing the nuclear charge.A third process for which reliable potential-energy curves are necessary is the ion-molecule (or ion-atom) reaction with charge exchange an example of which is shown in fig. 2. Relatively high accuracy is especially important since the various reaction channels are often very close in energy. The Na (,S)+Mg+ (,S)energy at 2.506 eV is only 0.2 eV CALCULATION OF MOLECULAR SPECTRA 14.0j 12.0 -10.0 -< 8.0: .4-6.0 4.0 -2.01-0.0 - ‘ “ ‘ I V “ A. t 5.0 10.0 15.0 Rlao Fig. 2. calculated potential-energycurves for various statesinNa+ +Mg (NaMg)+and Na +Mg+. The estimated full CI values are given. 2.5-2.0-.3 1.5-4 Q 1.0-0.5 -\ I 0.04 3.0 3:5 4.0 4.5 5.0 RscHIao Fig. 3. Calculated MRD-CI potential-energy curves for the low-lying states in ScH. below that of Na+ (lS)+Mg (3P),while the corresponding calculated data are 2.514 and 0.12 eV. Similar relations hold for the fourth and fifth reaction channels. The (Na++Mg) (”) curve crosses that of 3C+ and a charge-exchange reaction near the crossing point of both curves leads to a Na +Mg+ system.The potential curves agree very nicely with the experimental observation~.~~ S. D. PEYERIMHOFF I I I \ 0.4 I % 0.0 x311 A311 A31T a x" 'x31-\x3r 0.4 SiN Sip X2E+ x2x+ 0.0 CN CP Fig. 4. Calculated energy differences between the lowest states of isovalent compounds always evaluated for the respective equilibrium geometry. The last important feature to be mentioned in this section is the possibility of obtaining more insight into the electronic structure and an understanding of the causes of the behaviour of potential-energy curves; this in turn allows qualitative predictions to be made without resorting to additional calculations. Two examples illustrate this point. In transition-metal hydrides the magnitudes of the structural parameters are sometimes surprising.For example in ScH (fig. 3) the ground state XIC+ curve exhibits entirely different behaviour from the other multiplets of seemingly the same electronic configuration 6a2d17a1. Closer analysis shows that while this is the configuration for 371A(db7a) lII(dn7a) and 3C+(da7a) there is a second quite important occupation 39 7a2possible for the singlet states lC+(da7a and 7a2) with higher bonding properties and mixing of both configurations leads a marked shift of R,(X lC+) to smaller bond lengths and higher values of co,. Note also that the CI is very important in this example since the SCF approach alone places lC+ above %+ and would predict erroneously a 3A ground state for ScH. Ths measured absorption bands are between 17690 and 18350 cm-l (2.19-2.37 eV) and must certainly be attributed to the 2 W-X lC+ transition which we calculate at 2.10 eV.32 Similar situations have been observed in other transition-metal hydride~~~ as long as the dn7a configuration can be decomposed into dn-lda7a.Finally one of the most attractive theoretical studies is a comparison of compounds in which first-row constituents are replaced by those of the second or higher rows. The increased orbital stability of a(pa+pa) relative to n(pn+pn) upon replacement of a first- by a second- (or higher-) row atom causes a predictable trend in the relative stabilities of corresponding electronic states and hence absorption and emission properties. A typical example is presented in fig. 4 for the 2C+(a~4) and 211(a2n3)states of the isovalent series CN CP SIN and SIP in which the state with double population CALCULATION OF MOLECULAR SPECTRA of the 0 MO and only three electrons in n orbitals gains increased stability relative to %+.The equivalent trend is present in the corresponding ionic compounds or in the related C, CSi and Si whereby the first-row species favour X lZ+(n4) the compounds with atoms from both first- and second-row favour 31-I(0n3)and the second- (and presumably higher-) row molecules favour X 3C-(02n2)occupation. In a similar manner the increase in relative stability of n* (and n as its in-plane component) relative to n upon introduction of second-row atoms causes a decrease in the splitting of the n and n ionization potential (i.p.) in H,ABH so that n is the first i.p.in H,CN both i.p. are close together in H,SiNH and H,CPH while in H,SiPH+(2A”) (n-+ a)is the ground state. Equivalent arguments based on n n and n* orbital stability predict the location of Rydberg and (n n*) and (n,n*) states whereby it is of interest that only in the first-row compound H,CNH the 3(n,n*) is the first excited state as known from H,CO for example while already in the mixed compounds as well as in H,SiPH (if the geometrical H,ABH structure is assumed) the transition to 3(n,n*)is the lowest in the vertical region. FINE STRUCTURE The MRD-CI wavefunctions can also be used for the calculation of multiplet splitting. For this purpose the Hamiltonian is written in the form H = HO +H, +H, whereby the spin-orbit and spin-spin operators are taken from the Breit-Pauli formulation as All integrals are evaluated 37 Typical examples are the multiplet splittings for the X 3C-states of S, SO or 0,38 and the atomic states of the Br atom.39 If in SO the spin-orbit coupling between the 3C-state and the four b lX+ A” 3X+ C 311 and lll states is taken into account in the second-order perturbation sum Xi+k( i 1 H, 1 k),/AE a value for the zero-field splitting parameter D between 7.96 and 9.15 cm-l is obtained as long as A0 basis sets of double-zeta quality or better are employed.40 The contribution of H, is much smaller betizeen 0.66 and 0.715 cm-l depending on the A0 basis sets but evaluated in the single-cunfiguration treatment only.The resulting value D,,+D, of ca.9.5 cm-l must be compared with the measured splitting of 10.55 cm-l. It can be assumed that the error is caused by truncating the perturbation sum since a CI in which all symmetry-adapted functions generated by the CI for the lX+ 3Z+ and 3f117 states are coupled by the H, operator leads to an increase in D, of the order of 1-1.5 cm-l and thus places the theoretical D,,+D, in very good agreement with the measured splitting. Another example is the X splitting in OH. Recent MRD-CI calc~lations~~ in a standard DZ-A0 basis find a zero-field splitting parameter averaged over the zeroth vibrational level of A(v,) = -138.543cm-l compared with the measured A(v,) = -139.054cm-l if only the diagonal contribution is considered. Second-order effects (off-diagonal elements) are found to be very small and decrease the value of A by 0.086 cm-l if contributions from the first 2C+,,C-,4C-and 411 states are taken into account.The values differ only slightly from those obtained in a more expanded A0 a finding which is consistent with our experience that spin-orbit matrix S. D. PEYERIMHOFF elements (standard accuracy) do not require any particular A0 basis-set extension40 relative to what is usually employed in excited-state calculations. One definite advantage of the theoretical treatment is that it allows us to determine in a straightforward manner the dependence of the spin-orbit interaction or multiplet splitting as afunction of some geometrical parameter or normal coordinate information which is more difficult to extract experimentally.The considerable dependence of this quantity on the geometry has been demonstrated on various occasionsL6 and in most cases it has been found to be a direct consequence of a change in the character of the states (variation in CI expansion) under consideration. An interaction of spin-orbital momentum and rotation leads to additional splitting of the R+ and R-levels in the X 211 state of OH and is normally referred to as A doubling. This quantity can also be calculated by employing the spin-orbit matrix elements between X and the first excited 2X+ state of OH in addition to the rotational coupling matrix element (211ILJ 2C+). The actual splitting is conventionally given in terms of two parametersp and q.If purely theoretical quantities are employed (ie.the various matrix elements as well as the calculated rotational constant B) one obtains41 values of p = 0.2352 cm-l and q = -0.0381 cm-l compared with the experimental values of p = 0.2357 cm-l and q = -0.0391 cm-l.These examples should suffice to demonstrate that the theoretical approach to the determination of fine-structure effects in molecular spectra is quite successful; it is expected that a considerable number of applications in this area will follow in the future. VIBRATIONAL PATTERNS The total wavefunction for a given state consists of an electronic part treated and a function describing the nuclear motion. The vibrational wavefunctions are usually generated by solving the Schrodinger equation for nuclear motion in which the Born-Oppenheimer energy curves as described above serve as the potential term often approximated to some convenient analytical form.The calculation of vibrational levels in diatomics is very simple; polyatomics are usually treated only in the harmonic approximation and for small bending amplitudes but while this problem is tractable the results are only of limited significance especially if more than the lowest excited states are involved. The general problem is especially complicated as soon as (1) large-amplitude motion (2) coupling of modes and (3) the breakdown of the Born-Oppenheimer approximation because of coupling of electronic and vibrational motion are present. The first two problems have been treated for the CH chromophore in CD,H in a joint experimental and theoretical study.43 The vibrational spectrum has been measured between 900 and 12000 cm-l under high resolution and the vibrational dynamics has been treated by employing an effective spectroscopic Hamilt~nian~~ on a potential-energy surface constructed from > 279 data points.The results show semi-classical zero-point amplitudes of 15" and a bending amplitude of > 45" for the N = 4 and 5 levels of the CH chromophore quantum numbers (N = vs+ivb); they demonstrate clearly to what extent the domain of small-amplitude motion is diverged from and to what extent the Taylor series expansion of the potential is expected to lack convergence in low order. The analysis finds that the tridiagonal Fermi resonance causes strong coupling between the stretching and bending vibrations in the CH chromophore and leads to fast energy redistribution between these two motions; the picture of a pure local-mode stretching motion can thus not be maintained.Another intriguing problem which requires coupling of electronic and vibrational states is the Renner-Teller effect;45 a general survey of the problem has been given CALCULATION OF MOLECULAR SPECTRA by Jungen and Me~er~~ The electronically degenerate electronic states and D~xbury.~’ of linear molecules are split in the course of bending where again the large-amplitude motion is important. Ab initio w~rk~~-~~ has been undertaken for AH systems such as NH, PH, SHZ BH, AlH and SiHz in their X states for CH, NHZ and SiH in their excited ,Ag states and for HNO+ HNF and C,.50 The potential-energy curves show minima at bent geometries for both 211 (,A’ ,A/’)components in NH, PH and SHZ while the upper component is linear in the next three molecules men- tioned above.Two procedures have been employed so far in the ab initio treatments (a) a variational calculation of vibrational levels is carried out for each of the two CI (electronic) energy curves and the molecular Hamiltonian (T,-+H,) is then diagonalized in the basis consisting of products of the electronic and vibrational wavefunctions of both states or (b) the molecular Hamiltonian is diagonalized in a basis built from products of linear combinations of the electronic wavefunctions obtained in the Born-Oppenheimer approximation and eigenfunctions of a two- dimensional harmonic oscillator.It has been shown for NH that both methods give the same numerical One problem in the treatment is the choice of the Hamiltonian for nuclear motion. In the AH systems the most suitable coordinates seem to be polar coordinates p for bending and 4 for the phase angle of the molecular plane with respect to a space-fixed plane (‘rotation’). If coupling of stretching and bending motions is neglected (or treated approximately at a different stage) the most convenient form in ab initio work describing large-amplitude bending seems to be T,-= -qT(1 P) a2/ap2+T,(P) vaP+ &(PI a2/a4,+ T,(P)l 2 whereby the coefficient~~l can be expanded in polynomial form conveniently in the same basis as used for the potential energy.In the limit p -+ 0 this form reduces to the Hamiltonian of a two-dimensional harmonic oscillator Results of such a treatment (in large A0 basis sets includingf-type functions in some instances) give for example T,f,tt values [differences between the lowest Z(K = 0) vibronic level of the upper electronic state and the first lower state level which can combine with it i.e.us = 0 K” = 11 for those species which are known experimentally of 1 1 150 cm-l (v’ = 1 V” = 0) for NH compared with 11 126 cm-l experimentally 4145 cm-l (4194 cm-l if reassignment of vibrational labelling is made) for BH, 18 820 cm-l (18 276 cm-l) for PH and 18 620 cm-1 (1 8 520 cm-l) for SHZ with measured values given in parentheses.Furthermore the character of vibronic levels above the barrier to linearity agrees well with experimentally extracted information in the few cases for which the latter is available; the calculations describe the degree of mixing of wavefunctions while the experimental information catalogues the levels as belonging to the ground- or excited-state potential or as mixed levels. These two examples demonstrate that the description of a complicated vibrational pattern can be tackled for certain systems but it should also be clear that such treatments even though suitable potential-energy surfaces exist or could be generated are far from routine and considerable ground-work still needs to be done in this area. TRANSITION INTENSITIES AND RADIATIVE LIFETIMES The calculation of intensities for spin-allowedelectronic transitions is straightforward and requires only the use of the electronic and vibrational wavefunctions for the states involved.The radiative lifetime ziof an initial excited level is determined by the sum 7;' S. D. PEYERIMHOFF 73 of all transition probabilities (Einstein coefficients) A to all (final) levels of the lower state as = X A = +Z AE:f IRif12. f f If rotational levels are neglected and the wavefunction is written as a product function Y = Yexvof electronic and vibrational terms the matrix element R is Hence the calculation of intensities requires in addition to the knowledge of energy levels and vibrational wavefunctions (which are not even necessary if a Franck-Condon factor of 1 is assumed) evaluation of the electronic transition moment Re,,.between electronic surfaces (in which usually the dipole operator Me = Z,,er summed over all electrons is employed); it reduces to the calculation of the dipole moment (as a function of nuclear coordinates) if the same state is involved and only vibrational transitions are considered.There is furthermore a simple relation between the Einstein coefficients for emission Aif and absorption Bif.An alternative quantity often used is the oscillator strength also based on the electronic transition probability as fe = IRer,,,12AEete-,or fere,lvtv,t if oscillator strengths are evaluated for transitions between vibronic levels. There are numerous calculations for intensities in the literature.la9 52-56 A typical example for the intensity pattern in an infrared spectrum is given in fig.5 which shows the Einstein coefficients for Av = 1 in the X 211ground state of OH. It is obvious that the various state-of-the-art calculation^^^ give similar results and represent the actual situation quite well. Electronic transition moments have mostly been evaluated for diatomics quite often as a function of internuclear separation. Generally the agreement between measured and calculated data is within 10-30% or at least smaller than a factor of two for quantities related to lRe~,7~12. Such deviations are probably not alarming since the errors in the experimental results are in a similar range. In a few cases as for example in the A lll,-X lXi Phillips system of C, the uncertainties are of concern because accurate knowledge of the absorption oscillator strength is a prerequisite for the determination of interstellar C abundances from astronomical observations; similarly these f values enter the analysis of the rotational populating of interstellar C and are needed to determine the contribution of the line opacity of carbon stars.Experimental values for foo vary from 1.41 x to 3.9 x (with much smaller error bars in each instance) while the calculated values of Di~hoeck~~ and of Chabalow~ki~~ are both 2.7 x a value of 2.5 x lop3 is obtained in a less extensive ab initio calc~lation.~~ Even though the theoretical data are within the experimental range of results in this case and the curve as a function of the C-C separation is parallel in the theoretical treatment and most experiments lifetimes are very sensitive quantities so that the A lll,(u' = 4) level lives almost twice as long (ca.1 1 ps) according to two recent experiments than is predicted by the calc~lations.~~~ 53 The source ofthe discrepancies between the various treatments is not known and awaits further explanation. More interesting from a theoretical point of view is the calculation of spin-forbidden transitions. This has recently been undertaken for a number of states in 0 (a lAS b lXi c 'Xi A' 3Au and A "L) for the n2states in S, SO NF and NC1 (a IA and b IZ) the 311ustate in C1 as well as for various states in simple hydrides (NH and ASH). In this case not only the dipole transition operator but also magnetic dipole CALCULATION OF MOLECULAR SPECTRA 50 l3 I i i 40 i 112 I! j! 30 j j!//f I v) 1 .i .T .I I.f: 20 10 0 rIIr,,,j 12345678 V' Fig. 5. Einstein coefficients for the Av = 1 transitions in the X ground state of the OH molecule obtained from different treatments. The MC-SCF and CEPA results are taken from ref. (57). 1 MRD-CI ; 2 MC-SCF; 3 MC-SCF/CI ; 4 CEPA2; 5 experimental. and (sometimes) electric quadrupole operators are necessary so that the Einstein coefficients for spontaneous emission can be written in the form A = ga3(AE)3(l(flRI i)12+$a2I (flL+2SI i) 12+&a2(AE)2I (flR x RI i),) where a is the fine-structure constant. It is important to consider both terms in the magnetic dipole moment operator i.e.the interaction of the radiation field with the orbital angular momentum as well as with the spin. The total electronic wavefunction can be written in a perturbation expansion as @rn=#rn+ ak(Pk k#m whereby all zero-order solutions #k are large-scale MRD-CI wavefunctions. The contributions of the perturbing states of different symmetries arise form the spin-orbit interaction and the corresponding coefficients akare evaluated as ak = ((Pk Iffsol 4m)/ (&rn-&k). For the X "; a Idgand b lZ; states in 0 and S the following perturbing states have been for X "c the states 'Z; and 3v111g for a lAg the state 311g and for b 'Cg the states "c and 311g. The results are contained in table 1.First good agreement between the observed data for 0 and those obtained theoretically is seen. All values reported for S are predictions. Secondly in contrast to measurements the calculations allow an analysis of the transition mechanisms. It is found that the dominant term in the b 1Cg+-3Cg transition moment is the spin contribution (spin flip in ";) to the magnetic dipole operator b1Cg-3C; (m,= _+ 1) transitions corresponding to a much larger lifetime of 6 x lo6 s. The major term determining the lifetime of a lAg is the orbital angular moment term in the magnetic dipole operator and arises from lAg-lIIS and X3C,-311g transitions in the perturbed X3Cg and a lAS S. D. PEYERIMHOFF 75 Table 1. Calculated lifetime~~~-~l (in s) for the first two excited states in various molecules possessing the same electronic configuration.Experimental values whenever available are given in parentheses. 02 s2 so NF NC1 b lZ+ 11.65 11.0" 3.4 0.013 0.018 0.002 (12) -(0.007) (0.015 0.022) (0.0006) a lA 5270,4330" 350 0.45 3.13 0.8 (3900) -(1 5.6) (0.002) b lP-u lA 720 780 450 -428 (400 within a --factor of two) " Two states of each perturbing symmetry are included in a slightly different A0 basis and type of treatment. wavefunctions. The lifetimes in S are smaller since the spin-orbit matrix elements are considerably larger and hence the perturbers to the zero-order wavefunctions are more important. The b lC,'-a lAS transitions arise from quadrupole interaction and are thus independent of spin-orbit effects i.e.show approximately the same intensity in 0 and S (table 1). In the heteronuclear SO molecule two states of 39111 3Z-and lC+ symmetry have been included in the perturbation expansion. The lifetimes are considerably shorter because the absence of the inversion symmetry favours electric (rather than magnetic) dipole processes. The b lC+-X 3C-transition borrows its intensity predominantly from terms connecting b lZ+-2 lC+ and X 3Z-2 3Z- which occur as perturbers in the pure-spin wavefunctions. For a lA the calculations predict intensity borrowing from A 311-X 3Z-and C 311-X 3C-as well as lA-lll and lA-2 lII dipole transitions. Mechanisms in the other two heteroatomic systems which have been calculated in a comparable manner,61 are equivalent.Finally an interesting study6 is the absorption intensity of the 2 lIIUand 2 311u states in Cl,. All states up to 90000 cm-l have been allowed to couple via the spin-orbit process with the 31Tu state but it was found that only the 2 llTU state couples to any significant extent. The oscillator strength for the singlet X Xi-2 lIIu transition evaluated in the usual way as indicated is 0.07; the triplet is found to be weaker only by a factor of 8 in good accord with absorption experiments which estimate a factor between 5 and 9. In summary it is seen that calculations are able to predict lifetimes of excited states with good accuracy over a timescale of twelve powers of ten from to > lo3 s. This possibility together with the energy predictions makes modern CI-type calcu- lations a very valuable tool for spectroscopic studies with a very wide field of application.The author wishes to thank all her colleagues and associates who have helped obtain the results reported in this work in particular Prof. R. J. Buenker (Wuppertal) Dr M. PeriC (Beograd) and Drs P. J. Bruna C. M. Marian and R. Klotz (Bonn). The financial support given to this work by the Deutsche Forschungsgemeinschaft over the past 4 years in the framework of the Sonderforschungsbereich 42 (energy levels of small molecules quantum-mechanical and experimental investigations) has been essential to this study and is gratefully acknowledged. 76 CALCULATION OF MOLECULAR SPECTRA (a) R. J. Buenker and S. D. Peyerimhoff in New Horizons of Quantum Chemistry ed.P. 0.Lowdin and B. Pullmann (D. Reidel Dordrecht 1983) p. 183; (b) R. J. Buenker S. D. Peyerimhoff and W. Butscher Mol. Phys. 1983 35 771. R. J. Buenker and S. D. Peyerimhoff Theor. Chim. Acta 1974,35 33; 1975,39 217. (a) R. J. Buenker in Quantum Chemistry and Molecular Physics into the 80s ed. P. G. Burton (University of Wollongong Press Wollongong 1980),p. 1.5.1 ;(6) R. J. Buenker in Studies in Physical and Theoretical Chemistry ed. R. Carbo (Elsevier Amsterdam 1982) vol. 21 p. 17. N. C. Handy and R. J. Harrison personal communication. P. G. Burton R. J. Buenker P. J. Bruna and S. D. Peyerimhoff Chem. Phys. Lett. 1983 95 379. E. R. Davidson in The World of Quantum Chemistry ed. R. Daudel and B. Pullmann (D.Reidel Dordrect 1974) p. 17. ' P. J. Bruna S. D. Peyerimhoff and R. J. Buenker Chem. Phys. Lett. 1980 72 278. M. Lewerenz P. J. Bruna S. D. Peyerimhoff and R. J. Buenker J. Phys. B 1983 16 451 1. S. D. Peyerimhoff and R. J. Buenker in Computational Methods in Chemistry ed. J. Bargon IBM Research Symposia Series (Plenum Press New York 1980) p. 175. lo P. J. Bruna and S. D. Peyerimhoff Proc. 16th Znt. Symp. Free Radicals Belgium Bull. SOC. Chim. Belg. 1983 92 525. l1 P. J. Bruna G. Hirsch R. J. Buenker and S. D. Peyerimhoff in Molecular Ions Geometric and Electronic Structures ed. J. Berkowitz and K. 0.Groenefeld NATO AS1 Series B (Plenum Press 1983) vol. 90 p. 309. l2 F. P. Hubermann J. Mol. Spectrosc. 1966 20 29. l3 R. P. Tuckett and S. D. Peyerimhoff Chem.Phys. 1984,83 203. l4 S. D. Peyerimhoff H. Dohmann and C. M. Marian in Symp. Atomic and Surface Physics ed. F. Howorka W. Lindinger and T. D. Mark (STUDIA Studienforderungsges. mbH Innsbruck 1984). l5 P. J. Bruna G. Hirsch S. D. Peyerimhoff and R. J. Buenker Mol. Phys. 1981 42 875. l6 C. M. Marian R. Marian S. D. Peyerimhoff B. A. Hess R. J. Buenker and G. Seger Mol. Phys. 1980 40,1453. l7 M. Lewerenz P. J. Bruna S. D. Peyerimhoff and R. J. Buenker Mol. Phys. 1983,49 1. la M. Bettendorff R. J. Buenker S. D. Peyerimhoff and J. Romelt Z. Phys. A 1982 304,125. l9 M. Bettendorff S. D. Peyerimhoff and R. J. Buenker Chem. Phys. 1982,66,261. 2o D. M. Hirst and M. F. Guest Mol. Phys. 1980 41 1483. 21 E. F. van Dishoeck M. C. van Hemert and A. Dalgarno J.Chem. Phys. 1982 77 3693. 22 P. H. Cribb Chem. Phys. 1984 88 47. 23 S. D. Peyerimhoff and R. J. Buenker Chem. Phys. 1981 57 279. 24 (a) M. C. Castex J. Le Calve D. Haaks B. Jordan and G. Zimmerer Chem. Phys. Lett. 1980 70 106; (b)T. Moller B. Jordan P. Gurtler G. Zimmerer D. Haaks J. Le Clave and M. C. Castex Chem. Phys. 1983,76 295. 25 A. E. Douglas and F. R. Greening Can. J. Phys. 1979 57 1650. 26 F. Grein S. D. Peyerimhoff and R. J. Buenker to be published. 27 M. Lewerenz B. Nestmann P. J. Bruna and S. D. Peyerimhoff Theochem.,in press. 28 E. Illenberger H-K. Scheunemann and H. Baumgartel Chem. Phys. 1979 37 21; Ber. Bunsenges. Phys. Chem. 1978,82 1154. 29 G. J. Verhaart H. A. von Sprang and H. H. Brongersma Chem. Phys. 1980,51 389; 1978,34 161.30 B. Nestmann and S. D. Peyerimhoff J. Phys. B in press. 31 F. von Busch personal communication. 32 J. Anglada Ph.D. Thesis in preparation. 33 J. Anglada P. J. Bruna S. D. Peyerimhoff and R. J. Buenker J. Mol. Struct. 1983 93 299. 34 P. J. Bruna S. D. Peyerimhoff and R. J. Buenker J. Chem. Phjv. 1980 72 5437. 35 P. J. Bruna V. Krumbach and S. D. Peyerimhoff Can. J. Chem.,in press. 36 C. M. Marian Diplomarbeit (University of Bonn 1977). 37 P. Chandra and R. J. Buenker J. Chem. Phys. 1983,79 366; 358. 38 B. A. Hess C. M. Marian R. J. Buenker and S. D. Peyerimhoff Chem. Phys. 1982 71 79. 39 B. A. Hess P. Chandra and R. J. Buenker Mol. Phys. 1984 52 1177. 40 R. Klotz C. M. Marian S. D. Peyerimhoff B. A. Hessand R. J. Buenker Chem. Phys. 1983,76,367.41 R. de Vivie Diplomarbeit (University of Bonn 1984). 42 S. R. Langhoff and C. W. Kern in Modern Theoretical Chemistry ed. H. F. Schaefer I1 (Plenum Press New York 1977) vol. 4 p. 381. 43 S. D. Peyerimhoff M. Lewerenz and M. Quack Chem. Phys. Lett. 1984 109 563. 44 H. R. Dubal and M. Quack Chem. Phys. Lett. 1980,72 342. O5 R. Renner 2.Phys. 1934 92 172. 46 Ch. Jungen and A. J. Merer in Molecular Spectroscopy Modern Research ed. K. N. Rao (Academic Press New York 1964) vol. 2. S. D. PEYERIMHOFF 47 G. Duxbury in Molecular Spectroscopy (Billing & Sons Guildford 1975) vol. 3. 48 R. J. Buenker M. Perid S. D. Peyerimhoff and R. Marian Mol. Phys. 1981 43 987; M. Perid S. D. Peyerimhoff and R. J. Buenker Mol. Phys. 1983 49 379. 4s M.Perid S. D. Peyerimhoff and R. J. Buenker Can.J. Chem. 1981,59,1318;Chem. Phys. Lett. 1984 105,44. 50 M. Perid and J. Radid-Perid Chem. Phys. Lett. 1979 67 138. 51 M. Perid M. Mladenovid J. Fejzo C. M. Marian and P. J. Bruna Chem. Phys. Lett. 1982,88 1982; M. Perid and M. Krmar Bull. SOC. Chim. Beogard 1982 47 43; M. Perid S. D. Peyerimhoff and R. J. Buenker Can. J. Chem. 1983 61 2500. j2 C. F. Chabalowski S. D. Peyerimhoff and R. J. Buenker Chem. Phys. 1983,81 57. E. van Dishoeck Chem. Phys. 1983 77,277. 54 J. 0.Arnold and S. R. Langhoff J. Quant. Spectrosc. Radiat. Transfer 1977 19 461. j5 M. Larsson and P. E. Siegbahn J. Chem. Phys. 1983 79 2270. j6 W. J. Stevens M. M. Hessel P. J. Bertoncini and A. C. Wahl J. Chem. Phys. 1977 66 1477. 57 H.J. Werner and P. Rosmus J. Chem. Phys. 1980 73 2319. 58 B. Poilly J. M. Robbe J. Schamps and E. Roueff J. Phys. B 1983 16 437. j9 R. Klotz C. M. Marian S. D. Peyerimhoff B. A. He13 and R. J. Buenker Chern. Phys. 1984,89,223. '"R. Klotz Ph.D. Thesis (University of Bonn 1984). 61 M. Bettendorff Ph.D. Thesis (University of Bonn 1984). 62 F. Grein S. D. Peyerimhoff and R. J. Buenker Can. J. Phys. 1984 in press.

ISSN:0301-5696    

DOI:10.1039/FS9841900063

出版商:RSC    

年代:1984

数据来源: RSC

摘要:

Faruduy Symp. Chem. SOC.,1984 19 79-84 Ab Initio Hartree-Fock Calculations for Periodic Systems BY V. R. SAUNDERS S.E.R.C. Daresbury Laboratory Daresbury Warrington WA4 4AD Received 12th December 1984 The present status of the application of Hartree-Fock theory to periodic systems using gaussian basis sets is reviewed. The most important technical aspects are discussed in a language close to that of the molecular theoretician so that the method can be seen as similar to a molecular calculation but with a number of extra features both simplifying and complicating. The level of agreement currently achieved between theory and experiment for such data as lattice parameters X-ray structure factors and electron momentum distributions (Compton profiles) is illustrated.The considerable potential for exploiting a number of the ideas stemming from periodic systems in the context of calculations on large molecules is highlighted. The application of Hartree-Fock theory to periodic systems has a number of advances in the past five years. Broadly speaking the subject’s status is similar to that of molecular Hartree-Fock theory in the early seventies. Thus it is now possible to deal with unit cells containing up to perhaps six heavy atoms using a double- zeta-quality gaussian basis set with polarization functions in favourable cases. The level of numerical accuracy is perhaps a little less than could be obtained in molecular calculations the total energy/cell being usually only correct to 10-100 phartree with concomitant errors in other properties.The largest error is not however numerical but stems from the use of finite basis sets and the Hartree-Fock approximation itself as in molecular calculations. The vast experience now gained in molecular work has had a decisive influence on the progress of the periodic-system work. For example the selection of a suitable crystal basis set is almost invariably rooted in atomic or molecular experience whilst many of the highly advanced molecular methods have migrated to periodic systems rather successfully. The present paper will argue that the time is now ripe for migration of some solid-state techniques for application in large molecules. The present work attempts to cover three aspects (1) those elements of the theory representing an extension of the molecular case (2) applications of the theory and (3) approximation schemes which have been well tested in the rather convenient context of periodic systems and which will almost certainly prove useful in calculations on very large molecular systems.THEORETICAL AND METHODOLOGICAL CONSIDERATIONS We assume a working knowledge of molecular Hartree-Fock theory and delineate the most important differences for periodic systems. Complication 1. The periodic system is of infinite extent. This leads to overlap and Fock matrices whose dimension is in principle infinite and similarly infinite summations arise when evaluating coulomb and exchange operators. This circumstance forces us to consider 79 HARTREE-FOCK CALCULATIONS FOR PERIODIC SYSTEMS Approximation 1.If the differential overlap between two gaussian basis functions is less than a pre-set tolerance then all matrix elements involving that overlap distribution are assumed zero. Typically a threshold of lop6is used in a high-quality calculation. However a problem remains; if one overlap distribution is deemed finite an infinite number may be generated by periodic repetition. This problem is solved by SimpZi$cation 1. Suppose we single out a particular unit cell call it the ‘zero cell’. Then it is sufficient to consider only overlap distributions involving at least one basis function within the zero cell since all others (where neither basis function is in the zero cell) are simply translational replicants with identical overlap and Fock matrix elements.This fact coupled with our use of approximation 1 gives rise to only a finite set of matrix elements which need to be computed to define the problem. Complication 2. Although the number of overlap distributions that need be considered explicitly has been reduced to a finite number matrix elements involving these distributions are made up of an infinite number of terms. For example the nuclear attraction operator consists of all atoms in the crystal while the coulomb and exchange matrix elements consist of double infinite summations over the entire crystal basis set. This difficulty leads to two further approximations Approximation2. Exchange operator matrix elements are evaluated using1 truncation criteria such that overlap distributions sufficiently remote from the target overlap distribution are simply ignored.Each target overlap distribution can thus be thought of as having an exchange ‘sphere’ surrounding it the size of the sphere being dependent on the nature of the overlap distribution smaller for inner-shell orbitals and larger for valence orbitals. The radius of the exchange sphere is normally surprisingly small it being rare to have to include interactions beyond fourth-nearest . neighbours. In fact the rate of convergence is considerably better by perhaps an order of magnitude than would be expected from a consideration of the largest- valued exchange integrals which have been excluded from the sum presumably because exchange interactions tend to have random phase.This quasi-localness of the exchange operator is not too surprising in view of the success of local density functional the~ry.~ Approximation 3. A coulomb matrix element represents the energy of interaction of a given target overlap distribution with the total electron distribution of the crystal while the nuclear attraction matrix elements represent the energy of interaction with respect to all nuclei in the system. The convergence of such sums is notoriously slow and recipes based on a straightforward truncation will be doomed to prohibitively long explicit summations if adequate accuracy is to be maintained. Satisfactory procedures are normally based upon a two-step analysis (a) the series is first rearranged by grouping together subsets of the total charge distribution; the purpose is to improve the convergence properties of the rearranged series whose summation index corresponds as a rule to the direct lattice vectors ordered in increasing length and (b)in the evaluation of the new series a decreasing level of accuracy is adopted with increasing index.Extreme care must be taken because connection problems may arise when passing from a zone of given accuracy to the next less precise one. In our method2 the total crystal electron density is partitioned into subsets by means of the well known Mulliken population analysis such that each shell of atomic orbitals has associated with it an electron distribution. Each shell distribution is asymptotically exponentially decaying with distance squared (assuming a gaussian basis) and so may be thought of as effectively having a finite extent.Coulomb interactions between an V. R.SAUNDERS overlap distribution and a shell charge are then evaluated according to three different schemes. (a)At short range where the penetration of the two charge distributions is appreciable the interaction is evaluated exactly as a linear combination of two-electron integrals. (b) At intermediate range each shell charge is replaced by a linear combination of point monopoles dipoles etc. (up to 1 = 6 is feasible with our current code) the poles being located at the centroid of the shell. The overlap distribution is allowed to electrostatically interact with these poles and the method can be seen as a practical outlet for the procedure of Stone5 for analysing charge distributions.Of course for the method to be economical it is necessary to be able to evaluate high-order multipole integrals and even more importantly electrostatic field integrals (involving operators of the form &,/R2z+1) efficiently. Fortunately very convenient recursive methods are now available.2 As the range increases the order of pole included may be gradually decreased until a distance is reached where only monopoles need be included so defining the border of the ‘quantum zone’. (c) Beyond this all shell charges are grouped together to form atomic charges (including the nucleus) and are said to belong to the Madelung zone. The interaction of the overlap distribution with the Madelung charges is evaluated by Ewald’s method.6 At the centroid of each overlap distribution the Madelung potential and its first gradients (with respect to distance) are evaluated whence the matrix element can be computed as a linear combination of overlap and dipole moment integrals.It is unnecessary to proceed to quadratic or higher terms as these have been shown“ to be of quite negligible importance. It may be argued that the use of the Mulliken partition of the charge is arbitrary. A ‘weighted’ Mulliken scheme has also been tried and shown2 to give negligibly different results. This invariance to the partitioning scheme is only realized if the multipole expansions are carried out to sufficiently high order. The point is that any reasonable partitioning of charge will give satisfactory results if used consistently and carefully.However the use of different partitioning algorithms in different zones will give rise to very severe zone connection problems. Simplijication 2. Consider the short-range Coulomb interaction of an overlap distribution with shell charges. Within these summations will be found sub-sets of the form J(i,j) =Z (ijlkl;m) P(k1) m where the sum is over a number of distributions (kl;m)which differ only by translation vector m,and hence have the same density matrix element P(k1). During integral evaluation it is convenient to store such quantities summed over the m index (which will normally embrace between 10 and 100 terms). The advantages are twofold (a) much less file space is consumed by the presummed integrals1 and (b)methods2.are available for evaluating such sums much more efficiently than evaluating each integral separately and then summing. Similar advantage may also be gained when evaluating sums over exchange or electrostatic field operators. Complication 3. Although the Fock and overlap matrix are now defined in terms of a finite number of distinct elements they are nonetheless infinite in dimension because of periodic replication. Fortunately this final problem may be overcome by Simplijication 3. Symmetry adaption of the basis set using the infinite number of translational symmetry operators gives rise to the Bloch basis. The Fock and overlap matrices may be evaluated at a finite set of points in k space (typically 10-150 such points are used) each k-space sub-block of the matrices being of the dimension of HARTREE-FOCK CALCULATIONS FOR PERIODIC SYSTEMS the number of basis functions per unit cell.After diagonalization the real-space density matrix may be found by numerical integration8 over the k-space density matrices. For further discussion see ref. (1) and (9). In summary the procedure outlined above is based almost entirely on real-space considerations and the severity of the approximations decay with distance in a manner which is highly regular and predictable and so can be adequately controlled. The general format of the calculation looks rather like a typical ab initio large-cluster calculation with the addition of a Madelung potential and periodic boundary conditions.The latter are crucial in allowing a much larger quantum zone than could be tolerated in a finite-cluster study since often 200-1000 atoms are involved. This means that periodic systems form the ideal test-bed for approximation schemes and in our opinion would have been invented for this purpose even if they had not occurred in nature. It is well to contrast the present method with that of Harris and Monkhorst,lO who proposed to work more directly in reciprocal space using Fourier transforms of Slater-type orbitals. Sums over real lattice points are replaced by sums over the reciprocal lattice. The principal difficulty of this method appears to be one of defining adequately accurate truncation criteria particularly when one is denied the advantage of the insight provided by being able to think in real-space terms.We may also mention here that there seems to be little advantage for the present method in the use of pseudo-potentials for the treatment of inner-shell orbitals. For example in a minimal-basis calculation on silicon (diamond structure) only ca. $ of the integrals involve inner shells. The reason is that the classification adopted in the approximation schemes delineated above makes a severe distinction between compact inner and the more diffuse valence shells. SOME ILLUSTRATIVE RESULTS We first choose to illustrate the importance of including Madelung effects particularly in ionic systems. Consider MgO in a double-zeta basis the molecule gives rise to a Mulliken oxygen charge of 0.8e.If we now perform a crystal calculation using a quantum zone of radius 25 bohr (this is about as large as can usually be afforded) but ignoring the Madelung potential the oxygen charge increases to 1.2e. After inclusion of the Madelung potential the charge separation increases to 2e and the system is fully ionic. In the calculation which did not incorporate the Madelung potential there is other evidence that all is not well for example certain Fock matrix elements which should be equal (because of periodicity) are found to differ by 0.3 hartree; this symmetry is restored (to six decimal places) upon inclusion of the Madelung field. A detailed study3 of f.c.c. lithium hydride (containing the very polarizable H anion) reveals even greater surprises because here a choice of two different unit cells produced totally different (but wrong) results if the Madelung potential was neglected (for example the total energy/cell differed by 0.2 hartree) whilst agreement was restored (to six- decimal places) if the Madelung potential was incorporated to gradient terms.If the Madelung potential included only zero-order terms only marginal improvement occurs (the energy per cell differed by 0.05 hartree). These calculations gave an optimized nearest-neighbour distance of 3.876 bohr (experimental value 3.858 bohr) and a binding energy of 0.338 hartree (experimental value 0.346 hartree). The calculated bulk modulus was within the large experimental uncertainty but it is not yet clear that the calculations are carried out with sufficient precision to allow a definitive result for such second-gradient properties.The calculated electron-density distribution which corresponded closely to a fully ionic model gave excellent V. R. SAUNDERS agreement with respect to experimental X-ray structure factor and Compton profile data. In contrast calculationsll using a simple ‘molecular simulated crystal model ’ which corresponded to a more covalent situation gave equally good agreement for the structure-factor data but rather poor agreement with the Compton profiles indicating that Compton-profile data more powerfully discriminate between theoretical models of electronic structure. A similar study’ of Li,N (space roup p6/mmm) gave rise to optimized lattice parameters a = 3.61 A and c = 3.84 x (experimental values 3.655 and 3.874 A); Li,N layers alternate with pure Li layers in the c direction with the Li atoms of the Li,N layers arranged in graphitic hexagonal structure with the N atoms at the centre of the hexagons while the Li atoms of the pure Li layers are atop the N atoms.The calculated electronic structure was found to correspond closely to the fully ionic model with the N atom being triply negatively charged and again excellent agreement with both X-ray structure-factor data and Compton-profile data was observed. Simple model calculations give good agreement for structure-factor data13 but poor agreement for the Compton profiles,14 reinforcing our view that structure-factor data are not as useful as Compton-profile data in discriminating between theoretical models.The calculated binding energy was 0.196 hartree (experimental value 0.42 hartree) an error which can be almost entirely attributed to the difference in correlation energy of the N atom and the N3- ion which may be estimated from Clementi’s tables15 to be 0.22 hartree. The calculations were also interesting because a major attempt at basis-set optimization (particularly for the N anion) was attempted in the crystalline environment. We turn now to consider briefly some results for the (SN) linear polymer,16 which is an almost planar chain and has a repeat unit of the form with the NS bonds parallel to the fibre axis being slightly shorter than those approximately perpendicular (1.593 and 1.628 A respectively).The bond angles x and y are 106.2 and 119.9’ respectively. The calculations used a minimal basis with and without d orbitals on the S atom the basis being as used by Palmer and Findlay17 in finite-cluster calculations on the same system. The polymer-optimized d-orbital exponent was 0.32 not too different from that found in the (SN) molecular system.18 The contribution of the d orbitals is found to be even more important in stabilizing the polymer than in the case of the (SN) precursor to such an extent that the chain remains unstable if d orbitals are excluded the d functions providing 0.24 hartree per cell with a sulphur atom d-orbital population of 0.29e and a total sulphur positive charge of 0.46. The d orbitals play a considerable role in reducing the ionicity of the S-N bonds (the S atom charge is +0.63 in minimal basis).Indeed in minimal basis the S atom can only achieve the necessary hypervalency by becoming ionic a tendency much reduced when the extra mechanism for bonding provided by the d functions is allowed for. The convergence of the finite-cluster calculation^^^ appears to be rather good if one considers the total energy per cell. However this does not mean that other aspects of the electronic structure are so rapidly convergent; even at the centre of the largest cluster (SN),, the electron distribution exhibits considerable differences when compared with the polymer result. One must be careful in taking even very large-scale cluster calculations as representative of the infinite system.HARTREE-FOCK CALCULATIONS FOR PERIODIC SYSTEMS CONCLUSIONS A brief and not too technical description of the techniques used in the Hartree-Fock theory of periodic systems has been given and test calculations used to illustrate the level of agreement with experiment. There are a number of hints here for large-molecule Hartree-Fock theory. (a) As a corollary of the observed slow convergence of Madelung potential effects finite-cluster calculations where one replaces a periodic system of ions with a finite array of point charges should be carried out with extreme caution taking care to use a suitable unit cell as the basis for replication. For example consider the linear chain ...A2+B2-A2+B2-A2fB2-A2+B2-... where the most obvious unit is either A2+B2-or B2-A2+.However finite-cluster calculations based on these cells will give very slow convergence because of the large dipole moment of the repeat unit.A better choice is either A+B2-A+or B-A2+B- which have zero dipole. Indeed it is possible to construct quadrupole-free repeat units (even in the general case) which would be even better. (b) The rapid convergence of exchange operator summations observed in periodic systems is almost certainly mirrored in large molecules. (c) The treatment of long-range coulomb forces in large molecules using a distributed multipole representation of the electron distribution should be tried in large molecules. (d) The procedure for efficiently summing over large numbers of interactions (simplification 2 above) is not applicable in orthodox SCF calculations because the density matrix elements vary through the SCF cycles.However in the direct SCF method,lg where one recomputes the molecular integrals at each cycle the technique should apply ufortiori. C. Pisani and R. Dovesi Znt. J. Quunturn Chem. 1980 17 501. R. Dovesi C. Pisani C. Roetti and V. R. Saunders. Phys. Rrr. B 1983 28 5781. R. Dovesi C. Ermondi E. Ferrero C. Pisani and C. Roetti Phjls. Rev. B 1984 29 3591. J. C. Slater Phys. Rev. 1951 81 385 R. Gaspar Acta Phys. Hung. 1954 3 263; W. Kohn and L. J. Sham Phys. Rep. A 1965 140 1133. A. J. Stone Chem. Phys. Lett. 1981 83 233. ti P. P. Ewald Ann. Phys. (N.Y.) 1921 64 253; M. Tosi in Solid State Physics ed. F. Seitz and D.Turnbull (Academic Press New York 1964) vol. 16 p. 1; F. E. Harris in Theoretical Chemistry Adtlances and Perspectires ed. H. Eyring and D. Henderson (Academic Press London 1975) vol. l. p. 147. V. R. Saunders in Methods in Computational Molecular Physics ed. G. H. F. Diercksen and S. Wilson (Reidel Dordrecht 1983) p. 1. * G. Gilat and L. J. Raubenheimer Phys. Rev. 1966 144 390; G. Gilat Phys. Rev. B 1973 7 891 H. J. Monkhorst and J. D. Pack Phys. Reti. B 1976 13 5188; D. J. Chadi Phys. Rev. B 1977 16 1746; J. D. Pack and H. J. Monkhorst Phys. Rez;. B 1977 16 1748. J. M. Andre L. Gouverneur and G. Leary Znt. J. Quanfurn Chem.. 1967. 1 427; 451 ;J. M. Andre J. Chem. Phys. 1969 50 1536. lo F. E. Harris and H. J. Monkhorst Phys. Rec. Lett. 1969 23 1026.l1 B. I. Ramirez W. R. McIntire and R. L. Matcha J. Chem. Phys. 1977 66,373. l2 R. Dovesi. C. Pisani F. Ricca C. Roetti and V. R. Saunders Phys. Rec. B 1984 30,972. l3 H. Schulz and K. H. Schwarz Acta Crystallogr. Sect. A 1978 34,999. P. Pattison and J. R. Schneider Acta Crystallogr. Sect. A 1980 36,390. E. Clementi. J. Chem. Phys. 1963 38,2248. l6 R. Dovesi C. Pisani C. Roetti and V. R. Saunders J. Chem. Phys. 1984 81 2839. M. H. Palmer and R. H. Findlay J. Mol. Struct. (Theochem) 1983 92 373. R. C. Haddon S. R. Wassermann F. Wudl and G. R. J. Williams J. Am. Chem. SOC.,1980 102 6687. J. Almhof K. Faegri and K. Korsell J. Comput. Chem. 1982 3 385.

ISSN:0301-5696    

DOI:10.1039/FS9841900079

出版商:RSC    

年代:1984

数据来源: RSC

摘要:

Faraday Symp. Chem. SOC.,1984 19 85-95 Generalizations of the Multiconfigurational Time-dependen t Hartree-Fock Approach BY DANNY L. YEAGER* Chemistry Department Texas A and M University College Station Texas 77843 U.S.A. AND JEPPE OLSEN AND POUL J0RGENSEN Chemistry Department Aarhus University 8000 Aarhus C Denmark Received 29th August 1984 We extend and generalize the time-dependent Hartree-Fock (TDHF) and multiconfigura- tional time-dependent Hartree-Fock (MCTDHF) approaches so that electronic transition energies from non-singlet reference states to states of pure spin symmetry other than the reference state can be described. Initial calculations are presented for the Be atom using the (2~3s) 3Sstate and the (2s2) l,!3 state as reference states.The accuracy of the calculated excitation egergies is examined as a function of extending the active space of the MCSCF reference state. Close agreement between the excitation spectra as obtained from either of the reference states is obtained when inner-outer correlation effects are considered in the MCSCF reference state. Over the past decade direct calculations of excitation energies transition moments and second-order molecular properties have proved to be very Various formulations have been given for the direct calculation of excitation processes. Examples are the equation-of-motion (EOM) method3 and the two-particle Green’s function or polarization propagator appr~ach.~ The simplest direct approach is the random phase approximation (RPA) or time- dependent Hartree-Fock (TDHF) approach where a single configurational state is used as reference state.l? 5+ For closed-shell systems with little correlation TDHF singlet excitation energies are generally determined to within an accuracy of 10% from experiment.Triplet excitation spectra may be structurally incorrect (triplet instabilities).For a reference state of spin different from zero calculations of excitation energies to states of the same spin multiplicity as the reference state are often very unreliable using TDHF.?? * A general description of how to evaluate excitation energies to states of spin multiplicity different from the reference state has not previously been given when the reference state has total spin different from zero. For closed-shell systems perturbational extensions of the TDHF approximation have been de~cribed.~? Substantial improvement has been obtained for both the singlet and triplet excitation energies in a second-order extension of the TDHF approximation.Second-order excitation energies appear to be accurate to ca. 0.5 eV for non-highly correlated systems. Third-order TDHF calculations have been describeds but the accuracy of such calculations is not yet well established.1° Open-shell perturbational extensions of the TDHF approximation have been described.* However severe problems need to be solved before accurate open-shell perturbation TDHF calculations can be carried out routinely. An alternative way of improving the TDHF approximation is to use the multi- configurational TDHF (MCTDHF) approximation where a multiconfiguration self- 85 GENERALIZATIONS OF THE HARTREE-FOCK APPROACH consistent field (MCSCF) state is used as reference state?* For both closed- and open-shell singlet reference states singlet and triplet excitation energies have been accurately evaluated with relatively short reference-state configuration lists (10-1 02 configurati~ns).~~ For a reference state of spin different from zero excitation energies to states of the same spin as the reference state can be described with equal accuracy.In this paper we demonstrate how extensions of the TDHF and MCTDHF approximation may be derived so that excitation energies to states of pure spin symmetry different from the reference state can be evaluated even when the reference state has a total spin that is different from zero.We do this by modifying the simple triplet particle-hole excitation operators such that the modified particle-hole operators when operating on the reference state of pure spin symmetry give states of pure spin symmetry. In electron propagator calculations on states of total spin different from zero annihilation and creation operators have previously been modified in a similarl43l5 manner to that used for the modification of the particle-hole excitation operators in this paper. This was done to assure that the ionization potentials and electron affinities of an electron propagator calculation are to states of pure spin symmetry. Initial calculation^^^^ l6on N2,0,and F with this approach [knownas the multiconfigurational electron propagator method (MCEP)] have yielded accurate ionization potentials for both principal and shake-up ionizations.In this paper we report results for beryllium and demonstrate for the (2~3s)3S+lS transitions that electronic transitions to pure spin states from a degenerate spin reference state may be accurately determined with the generalized multiconfigurational time-dependen t Hartree-Fock approach (GMCTDH F). THEORY SPECTRAL REPRESENTATION OF THE POLARIZATION PROPAGATOR The two-particle Green's function in the energy representation is defined using the Zubarev notation17 as ((A; B))E = (A+I (Ef+A)-' I B) (1) where A and B are number conserving operators and fand H are the superoperator identity and the superoperator Hamiltonian respectively.These are defined as !A = A (2) HA = [H,A]. (3) The Hamiltonian H is given as U au' where the creation and annihilation operators are indexed by a formally complete orthonormal basis of spatial orbitals and a spin index 0 which has the value of a or p spin. The superoperator binary product in eqn (1) is defined as (AI B) = <Wt" I [A+,4I Wt") (5) * It is apparently frequently forgotten that an AGP state may be thought of as a type of MCSCF state. AGP reference states usually do not obey the 'killer' condition and many calculations with AGP reference states arbitrarily zero the B matrix to force the 'killer' condition to be fulfilled. For theoretical and calculational analyses see for example ref.(1 2). D. L. YEAGER J. OLSEN AND P. JORGENSEN where Ity,"") refers to the exact N-electron reference state of total spin S and S component M. The spectral resolution of eqn (1) is (V,"" IA IV:'"') (VZ"' IB IV,"") -c E+ ~f-E:' nS'M' (V,"" 1 B IV:'"') (V:'"' IA I VfM> +c E+ E:'-~f nS'M' where the summations in the first and second terms contain a complete set of N-electronic eigenstates Eqn (6) shows that the poles of the two-particle Green's function occur at the excitation energies (E$-Ef)of the molecular system. The residues give the transition amplitudes (tyfMIA Iv/:'"') (~2'"' IB Iv/fM) which for example express the electric dipole transition probabilities if A and B refer to the electric dipole moment operator.APPROXIMATE CALCULATIONS The spectral representation in eqn (6)exhibits the physical content of the two-particle Green's function but it gives little insight into how approximate two-particle Green's function calculations can be carried out. To bring the Green's function to a form which is efficient for describing approximate Green's function calculations we take the inner projection1* of the superoperator resolvent in eqn (1) ((A; B))E =(B+Ih)(hIEf+ H Ih)-l (h1 A). (8) If the projection manifold h is complete eqn (8) and (1) are identical. Approximate two-particle Green's functions are determined by truncating the projection manifold and using an approximate state as the reference state. Below we describe in more detail how to carry out such approximate calculations.Excitation energies occur at the poles of the Green's function and may be determined according to eqn (8) as eigenvalues of the matrix equation (hIEf+ I? Ih) =0. (9) Eqn (9) is the same as the equations-of-motion method (EOM) for excitation energieslg <Vf" I[soh[K0R11 I V,"") =%(VfM I Wi,011IVf") (10) when the excitation operator 01is expanded in the projection manifold h and coAis the excitation energy. In practical calculations with either eqn (9) or (10) we will use the symmetric double commutator [A CI =:([A,[B'Cll+ "A BI Cl) (1 1) if it is not specified otherwise. EXCITATION ENERGIES FOR A NON-TOTALLY SYMMETRIC REFERENCE STATE The excitation energies E:' -E,S appearing in the spectral representation of the two-particle Green's function in eqn (7) are between states of pure spatial and spin symmetry.This was obtained because the complete set of states in eqn (6) was assumed to be eigenstates of the non-relativistic Hamiltonian in the Born-Oppenheimer approximation. In approximate two-particle Green's function calculations special GENERALIZATIONS OF THE HARTREE-FOCK APPROACH precautions must be taken to assure that the excitation energies have pure spatial and spin symmetry. From eqn (6) it is clear that the conditions necessary for obtaining excitation energies of pure spatial and spin symmetry consist of requiring (1) that the approxi- mate reference state has pure spatial and spin symmetry and (2) that the approximate projection manifold is constructed such that when operating on the approximate reference state states of pure spatial and spin symmetry are created.The first condition is trivial to satisfy. The second condition while easily fulfilled for a totally symmetric reference state is by no means trivially fulfilled for a non-totally symmetric reference state. To illustrate this point consider for example a reference state of total spin S and spin projection M I rySM). The singlet particle-hole tensor operator (for convenience the MCSCF orbitals are arranged in the order core valence and then the virtual orbitals) when operating on I rySM) will automatically create states of total spin S and spin projection M. However if one component of the triplet spin tensor particle-hole operators operates on I vSM) states of mixed total spin S+ 1 S or S-1 are formed.Excitation to states of mixed spin symmetry will therefore be obtained if the operators in eqn (13) are used in an approximate projection manifold. In the following we describe how to modify the operators in eqn (13) to ensure that excitation energies in an approximate Green's function calculation will be of pure spin symmetry. States of pure total spin S+ 1 S or S-1 can be formed by coupling the operators of eqn (13) with the various components of the reference state using appropriate vector coupling coefficients. For example a state of total spin S+ 1 and Sz component M may be formed as where the analytical expressions for the vector coupling coefficients given for example in Tinkham20 have been used.When the identities D. L. YEAGER J. OLSEN AND P. J0RGENSEN are inserted into eqn (14) we obtain (S-M+1) ‘2 +((S+M+ 1) ) apsa s+]VSM). I From eqn (1 7) it is obvious that modified (and unnormalized) particle-hole operators T,+,(1,O) may be defined as TFS(1,O) = -(S+ M+ 1) a& aspS-+(S+M + 1) (S-M + 1) (a:a asa -a& asp)+(S-M + 1) a asaS+. (18) These operators when acting on a state of totai spin S and S component M ( I SM)) give (unnormalized) states of total spin S+ 1 and S component M ( I S+ 1 M)). Modified particle-hole operators TTS(O,0) and T,+,(-1,O) which form states of total spin S and S-1 respectively when acting on 1 SM) may be derived in a similar manner as TFs(O,0) = a& aspS-+M(a&asa-a& asp)+a$ asaS+ (19) T:s(-1,O) = -(S-M)a&aspS--(S-M)(S+ M)(a& asa-a& asp)+(S+ M)a:p as&S+.(20) Use of the modified particle-hole operators of eqn (18)-(20) in an approximate projection manifold ensures that the excitation energies of an approximate Green’s function calculation are between states of pure spin symmetry provided the reference state has pure spin symmetry. MULTICONFIGURATIONALTIME-DEPENDENT HARTREE-FOCK (MCTDHF) We now discuss the multiconfigurational time-dependent Hartree-Fock (MCTDHF) approximation. The reference state is chosen as a multiconfigurational self-consistent field (MCSCF) state I OSM).21In most of our subsequent discussions we use a complete active space (CAS) MCSCF reference state. This choice is made to simplify evaluation of certain matrix elements by avoiding the calculation of three-body density matrices.To set up the projection manifold we also define the MCSCF orthogonal complement of states to JOSM) denoted (ITS’M’)}.For a CAS MCSCF state the orthogonal complement space { I TS’M’)} is here considered to be all linearly independent N-electron states of all possible symmetries of the CAS space orthogonal to J OSM). The projection manifold of the MCTDHF approximation consists of the state- transfer excitation R+and de-excitation R operators where ITSM) is a state in the orthogonal complement MCSCF space and the non-redundant set2’ of spatial totally symmetric particle-hole [Q+(OO) of eqn (12)] and hole-particle Q(0,O) excitation operators.The approximation projection manifold of the MCTDHF approximation may thus be written as h = {Q+<o,01 WO O) Q(0,O) R(O,0)). (22) 90 GENERALIZATIONS OF THE HARTREE-FOCK APPROACH Inserting this projection manifold into eqn (9) and using the MCSCF state I OSM) as reference state gives the MCTDHF approximation for excitation energies to states of the same spin multiplicity as the MCSCF reference state provided the superoperator Hamiltonian is defined to operate always first on the orbital excitation operators in evaluating coupling elements between orbital and state excitation operators.ll EXTENSIONS OF MULTICONFIGURATIONAL TIME-DEPENDENT HARTREE-FOCK When the reference state is totally symmetric in spin space (S = 0) an extension of the MCTDHF approximation may straightforwardly be defined for determining excitation energies to states of triplet spin symmetry.The projection manifold then may be chosen to consist of the simple particle-hole and hole-particle triplet excitation operators of eqn (1 3) and appropriate state-transfer excitation and de-excitation operators. For a reference state which is not totally symmetric in spin space care must be taken as described previously in the selection of the projection manifold to assure that the excitation energies are to states of pure spin symmetry. For a reference state of total spin S the projection manifold for excitation energies to states of total spin S+ 1 may be chosen to consist of where the T+ and T operators are defined in eqn (18).Excitation energies to states of total spin S-1 may similarly be obtained by using the projection manifold h(-1,O) = { T+(-1 O) R+(-1 O) T(-1,O) R(-1,O)). (24) A straightforward extension of the MCTDHF approximation for excitation energies to states of total spin S can also be defined by using the projection manifold The inclusion of the P(0,O)and T(0,O)operators relative to the MCTDHF manifold is justified because these excitation operators when operating on the reference state I OSM)create states that are single excitation in the spatial part of the wavefunction but with an additional spin flip. Such states are of approximately the same energy as the states formed by the simple orbital excitation Q+(O,0) and de-excitation Q(0,O) operators. Of course for a reference state which is totally symmetric in spin space (S= 0) all contributions from the l"t(0,O)and T(0,O)operators vanish.Furthermore care must be taken to assure that operators in eqn (25) are linearly independent. GENERALIZED MULTICONFIGURATION HARTREE-FOCK EIGENVALUE EQUATION Using the MCSCF reference state 1 OSM) and the projection manifolds of eqn (23F(25) allows us to define extensions of the MCTDHF approximation for determining excitation energies to states of total spin S+ 1 S and S-1 respectively. A generalized MCTDHF eigenvalue equation is obtained from eqn (9) As an example consider the case where excitation energies are determined to states 91 D. L. YEAGER J. OLSEN AND P. J0RGENSEN of total spin S+ 1. The matrices A(S+ 1,M) and S(S+ 1,M) of eqn (26) are then defined as A@+ 1 M) (OSM I[T(1,O) H W1,O)l I OSM)(OSM I “T(1,O) HI,R+(1 011 I OSM) =((OSMI[R(I,O),[H 1-~(1,0)111~S~)(~~~I[R(l,O),H,R+(1,0)11~SM) (27) S(S+ 1,M) (OSM I [T(l,O) P(1,O)l I OSM)(OSM IIT(L O) R+(L011 I OSM)).(28) =((OSMI [R(l,O) W1,O)I I OSM)(OSMI [R(1,O),R+(l,O)I I OSM) The matrices B(S+ 1,M) and A(S+ 1,M) are obtained from A(S+ 1,M) and S(S+ 1,M) by replacing T(1,O) and R(1,O) with T+(1,O) and R+(1 0) respectively. Note that in defining coupling elements between the orbital and state excitation operators the matrices A(S+ 1,M) and B(S+ 1,M) are defined such that the Hamil- tonian is commuted first with the orbital excitation operator and then the result is commuted with the state-transfer excitation operator.Matrix elements of A and B which refer exclusively to either the orbital or the state space are defined in terms of the symmetric double commutator. The matrix elements that are required to determine excitations to states of total spin S or S-1 are defined in a similar manner. In ref. (2)we give the explicit expressions for the matrix elements of the A B S and A matrices corresponding to excitations to states of total spin S-1 S or S+ 1 for a CAS MCSCF reference state. A CAS MCSCF state is considered because the evaluation of A and B simplifies in this case. Only two-electron density matrix elements are required in evaluating A and B for a CAS MCSCF reference state while for a general MCSCF reference state three-electron density matrix elements may also have to be constructed.Straightforward evaluation of the matrix elements of the A and B matrices requires knowledge of one- and two-electron density matrices with different M components of the reference state. Using the Wigner-Eckart theorem the expressions for the A and B matrix elements may be simplified to only require knowledge of one- and two-electron density matrix elements with the same M component of the reference state. To illustrate both these points we consider the evaluation of one of the right-hand side components of the A matrix (SMI[Q+(l -I>S+,[H,Q+ (1 -l)~+lllSM) =2/((S+ M+ 1)(S+ M+ 2)(S-M)(S-M-1)) x (SMI [Q+<l,-I) [H,Q+(l -I)]] I SM+2) +2/(2(S+M+l)(S-M))(SMI(Q+<l -1) x [H,Q+<1,0)l-[H,Q+(l -1)1 Q+(l 0))I SM+ 1).(29) Consider initially how to evaluate the first term of eqn (29).The operator inside the bracket is a tensor operator of rank 2 and component -2 p-2 = [Q(l -I) [H,Q+(l -I)]]-(30) Using the Wigner-Eckart theorem we obtain (SMI VySM+2) = (S(M+2)2-2(SM)(SII VIIS) (31) 92 GENERALIZATIONS OF THE HARTREE-FOCK APPROACH where (S(M+2)2 -2 ISM) is the vector coupling coefficient and (SI I v2 I I S) is the reduced matrix element. To evaluate the first term in eqn (29) we need thus to evaluate the reduced matrix element (S(I v2 I IS). Using eqn (30) we determine the tensor operator q as 1 1 v2 -[S+ ?el] =-[S+ [S+,P-,I] "~'6 2d6 The reduced matrix element may therefore be determined from and the first term in eqn (29) may be evaluated as (SMI [Q+(l -11 [H,Q+<l,-I)]] I SM+2) +2[Q'(l 01 [H,Q+(l,O)ll+ Q+(l -11 [H,Q+<l 1)11> ISM).(34) Eqn (34) requires only one- and two-electron density matrices referring to the state ISM) to be evaluated. For a general MCSCF state the second term in eqn (29) will contain three-electron density matrix elements. For a CAS MCSCF however where orbital excitation operators within the active space are redundant we have defined the ordering of orbitals so that QrsISM) = 0. [See the discussion following eqn (12).] We may therefore write the second term in eqn (29) as a double commutator d(2(S+ M+ 1)(S-M))(SM I {[Q+(l,-11 [H,Q+(l O)]] +[Q+(l,o) [H,Q+(l -1)11>I SM+ 1). (35) It requires then that only one- and two-electron density matrix elements be evaluated.Using that we may write eqn (35) as 2d((S+M+ 1) (S-M))(SM I P1I SM+ 1) = 22/((S+ M+ 1)(S-M))(S(M+ 1)2 -1ISM) (SII PII S) (37) and using eqn (31) the second term in eqn (29) may be easily evaluated with only knowledge of one- and two-electron density matrix elements involving the state I SM). RESULTS AND DISCUSSION We report now some initial calculations of the excitation energies of the Be atom using the TDHF MCTDHF and generalized MCTDHF schemes outlined above. First we use the (2s2)lS state as reference state and calculate excitation energies to states of lS and 3S symmetry. The calculated MCTDHF excitation energies are D. L. YEAGER J. OLSEN AND P. J0RGENSEN Table 1. Excitation energies to states of 'S and 3S symmetry in Be (29) 1s -+ (2sns) ' 3s (2~3s)3S+ (2sns) 'l3S state expa 2sb 2s2pb 2~2~~2~2~'~2s2s'2p2p'3db 2s3sb- 2~2~'3~3~'2p2p'~* (2s2) 1s (2~3s)'S (2~4s)lS (2~5s)'S (2~6s)'S 0 6.78 8.09 8.59 8.84 0 6.12 7.26 7.74 8.98 0 6.95 8.26 8.79 9.17 0 6.76 8.07 8.60 8.98 0 6.77 8.08 8.60 8.98 0 5.67 6.51 7.03 7.44 0 6.74 8.22 8.64 10.23 (2~3s)3S (2~4s)3S (2x5s) 3S (2~6s) 6.46 8.00 8.56 8.82 5.49 7.10 7.67 8.88 6.50 8.15 8.74 9.06 6.43 7.98 8.56 8.88 6.43 7.98 8.56 8.88 5.00 6.44 7.00 7.34 6.39 7.93 8.50 8.83 a S.Bashkin and J. 0.Stoner Jr Atomic Energy Levels and Grotrian Diagrams I Addenda (North-Holland Amsterdam 1978). Complete active space in specified orbitals. Entries in this column were obtained by adding the calculated (2~3s) 3S -,(2s2)lS excitation energy of 5.00 eV to the obtained excitation energies.Entries in this column were obtained by adding the calculated (2~3s)3S -+ (2s2)'S excitation energy of 6.39 eV to the obtained excitation energy. examined as a function of the quality of the CAS space in the MCSCF reference state. The excitation energies are then calculated using the (2x3s) 3S state as reference state and finally the quality of the excitation spectra using the (2s2)?S state and the (2~3s) 3S state as reference states are compared. The basis set we use is a 5s contraction22 of a H~zinaga~~ 10s Gaussian basis set five sets of uncontracted Gaussian p functions24 and a set of Gaussian dfunctions with exponent 0.65. Four additional diffuse s functions with exponents (0.020,0.008,0.003 0.001) four additional sets of diffuse Gaussian p functions with exponents (0.016 0.007,0.003,0.001) and two sets of diffuse Gaussian dfunctions with exponents (0.015 0.0032) were added.In table 1 the obtained excitation energies are reported. In the calculations of the third column the active space consists of the 2s orbital. The reference state therefore is the SCF (2s2)lSstate and the excitation energies become those of the TDHF approximation. The excitation energies of Be have previously been calculated in the TDHF approximation using a 50 Slater-type orbital basis. The excitation energies of both calculations are essentially identical indicating that the reported TDHF excitation energies must be close to the complete basis limit. In the fourth column the active space is extended with the 2p orbital.The MCSCF reference state therefore contains two configurations (2s2 and 2p2). MCTDHF excitation energies have previously been reported in ref. (13) using the same two- configuration reference state and with a 50 Slater-type orbital basis. The lowest two excitation energies of both singlet and triplet symmetry are identical to all reported values in the two calculations indicating that these excitation energies must be close to the complete basis limit for the two-configuration case. In column 5 we report the MCTDHF excitation energies that are obtained by considering the so-called inner-outer correlation effects in the MCSCF reference state. The results of column 5 show the importance of inner-outer correlation of the MCSCF reference state.In column 6 we have further included the angular correlation effects which originate from GENERALIZATIONS OF THE HARTREE-FOCK APPROACH including a 3d orbital in the MCSCF reference state. The MCTDHF excitation energies do not change by including the 3d orbital in the active space indicating that the MCTDHF excitation energies are stable toward extending the active space with orbitals of higher angular momentum. The TDHF excitation energies are generally of rather poor quality and differ from experiment by as much as 0.5-1.1 eV. A substantial improvement is obtained when the reference state is correlated with the 2p2configuration. The MCTDHF excitation energies (column 5) then differ from experiment by 0.1-0.2 eV.When the inner-outer correlation is included in the MCSCF reference state agreement with experiment to within 0.01-0.03 eV is obtained for the lowest three excitation energies of both singlet and triplet symmetry. The larger deviation in the excitation energies to states originating from the 2s6s electronic configuration is due to the fact that our basis only contains 9s Gaussian basis functions and therefore cannot accurately describe more than the few lowest excitation energies. In columns 7 and 8 we report the excitation energies obtained using the (2~3~)~s state as reference state. In column 7the 2s and 3s orbitals are active and the reference state becomes the (2~3s) 3SSCF state. The TDHF excitation energies of column 7 are very unreliable for excitations of triplet-triplet and triplet-singlet character.Note that for the SCF reference state (2~3s) 3S we have the option when evaluating excitation energies to states of ‘5 symmetry either to include the 2s+ 3s orbital excitation operator in the calculation or to include the state-transfer operators containing the 2s2,3s2and 2s3s singlet states. The results in column 7 use the state-transfer operators as only this choice gives the possibility of describing all the excitation energies in table 1 in a THDF approach. In column 8 we report the excitation energies which are obtained by including the 2s2s’3s3s’2p2p’orbitals in the active space. This active space describes both inner-outer correlation and some angular correlation in the MCSCF reference state.As is clear from the results in column 8 very good agreement is obtained with experi- ment for the lowest 3 excitation energies of both triplet-triplet and the triplet-singlet character. The triplet-triplet excitation energies are evaluated using the MCTDHF approach. Inclusion of the T operators in the projection manifold [see eqn (25)]gives no significant improvement and introduces in some cases linear dependencies in the projection manifold. The inclusion of the T operator has not been explored further for triplet-triplet excitations. The triplet-singlet excitation energies in column 8 have been evaluated using the projection manifold of eqn (24). The excitation spectrum in column 8 which uses the (2~3s) 3S state as reference state and the spectrum in column 5 which uses the (2s2)lSstate as reference state are in very close agreement and agree with experimental excitation energies.Excitation spectra may therefore equally reliably be evaluated using the ground or an excited state as reference state. It also appears that excitation energies involving a change in the total spin may be evaluated as reliably as excitation energies which conserve the total spin. Of course many more calculations have to be carried out before definite conclusions can be made about this point. D.L.Y. thanks the National Science Foundation (grant no. CHE-8023352) and the Robert A. Welch Foundation (grant no. A-770) for support and NATO (grant no. RG 193.80) for travel support. D. L. YEAGER J.OLSEN AND P. J0RGENSEN 95 J. Linderberg and Y. Ohm Propagators in Quantum Chemistry (Academic Press London 1973). J. Oddershede P. Jsrgensen and D. L. Yeager Comput. Phys. Rep. in press. C. W. McCurdy T. Rescigno D. L. Yeager and V. McKoy in Methods of Electronic Structure ed. H. F. Schaefer 111 (Plenum Press New York 1977) vol. 3 pp. 339-386. P. Jsrgensen Annu. Rev. Phys. Chem. 1975 26 359; J. Oddershede Ado. Quantum Chem. 1978 11 257. M. A. Ball and A. D. McLachlan Mol. Phys. 1964 7 501 ; A. D. McLachlan and M. A. Ball Rev. Mod. Phys. 1964 36 844. T. H. Dunning and V. McKoy J. Chem. Phys. 1967,47 1735. D. L. Yeager and V. McKoy J. Chem. Phys. 1975 63,4861. P. Swanstrsm and P. Jsrgensen J. Chem. Phys. 1979 71 4652. J. Oddershede and P. Jsrgensen J.Chem. Phys. 1977 66 1541. lo D. L. Yeager and K. F. Freed Chem. Phys. 1977 22 415. l1 D. L. Yeager and P. Jsrgensen Chem. Phys Lett. 1979 65 77; E. Dalgaard J. Chem. Phys. 1980 72 816. l2 J. Linderberg and Y. Ohm Int. J. Quantum Chem. 1977 12 161; Y. Ohm and J. Linderberg Int. J. Quantum Chem. 1979,15,343; B. Weiner and 0.Goscinski Znt. J. Quantum Chem. 1977,12,299; B. Weiner H. J. Aa. Jensen and Y. Ohm J. Chem. Phys. 1984,80 2009. Is D. L. Yeager and P. Jsrgensen Chem. Phys. Lett. 1979 65 77; P. Albertsen P. Jsrgensen and D. L. Yeager Chem. Phys. Lett. 1980 76 354; P. Albertsen P. Jsrgensen and D. L. Yeager Mol. Phys. 1980,41,409; P. Albertsen P. Jsrgensen and D. L. Yeager Int. J. Quantum Chem. Symp. 1980 14 229; D. L. Yeager J. Olsen and P.Jsrgensen Int. J. Quantum Chem. Symp. 1981 15 151; J. Nichols and D. L. Yeager Chem. Phys. Lett. 1981,84,77;D. Lynch M. Herman and D. L. Yeager Chem. Phys. 1982 64 69; P. Jsrgensen P. Swanstrsm D. Yeager and J. Olsen Int. J. Quantum Chem. 1983 23 959. l4 €3. T. Pickup and A. Mukhopadhyay Chem. Phys. Lett. 1981,79 109. l5 J. A. Nichols D. L. Yeager and P. Jsrgensen J. Chem. Phys. 1984 80 293. l6 €3. Thies J. Golab D. L. Yeager and J. Nichols to be published. l7 D. N. Zubarev Sou. Phys. Usp. 1960 3 320. P-0. Lowdin Phys. Rev. A 1965 139 357. D. J. Rowe Nuclear Collectiue Motion (Methuen London 1970). 2o M. Tinkham Group Theory and Quantum Mechanics (McGraw-Hill New York 1964). 21 J. Olsen D. L. Yeager and P. Jsrgensen Ado. Chem. Phys. 1983 54 1. 22 T. Dunning J. Chem. Phys. 1971 55 716. 23 S. Huzinaga J. Chem. Phys. 1965 42 1293. 24 T. Dunning and P. J. Hay in Methods of Electronic Structure ed. H. F. Schaefer 111 (Plenum Press New York 1977) vol. 3 pp. 1-28.

ISSN:0301-5696    

DOI:10.1039/FS9841900085

出版商:RSC    

年代:1984

数据来源: RSC

摘要:

Faraday Symp. Chem. SOC.,1984 19 97-107 Current Status of the Multiconfiguration-Configuration Interaction (MC-CI) Method as Applied to Molecules Containing Transi tion-metal At oms BY PERE. M. SIEGBAHN Institute of Theoretical Physics University of Stockholm S-11346 Stockholm Sweden Received 14th August 1984 The current status of the MC-CI method is reviewed. Calculations on transition metals show that the MCSCF step is often the most problematic part of these types of calculations. The reason is that the most desirable MCSCF calculation can easily be too large for presently available methods. The limitation lies in the number of configurations which can be handled. A new direct CASSCF CI method is suggested which is capable of efficiently handling a larger number of configurations.In the multi-reference CI step the limitation is often in the size of the reference space rather than the total number of configurations. Methods are therefore needed in which the internal space is more efficiently handled particularly with respect to the number of formulae which need to be stored. The new CASSCF CI method can also be useful in this context. 1. INTRODUCTION In the multiconfigurationsonfiguration interaction (MC-CI) method the zeroth- order subspace is selected by an initial MCSCF calculation. The most important configurations in the MCSCF calculation are then selected as reference states for a final singles and doubles CI calculation. In this paper the current status of the MC-CI method will be described.Examples of the models used and problems encountered in applications to molecules containing atoms of the second and third rows of the periodic table will be given. Based on these calculations possible directions of future developments of the method are discussed. It is of interest in this context to review the MCSCF and CI methods. This history which is given in the first section of the paper will of necessity be very brief and incomplete. Only some of the most important references will be mentioned. Since the ideas behind the MCSCF and CI methods are very simple they were suggested soon after the development of quantum mechanics in the early 1930s. Owing to the lack of sufficiently powerful computers the applications of these methods to molecules did not appear until the end of the 1960s.The first applications were naturally on diatomic molecules. The early MCSCF calculations had a CI expansion of only a few terms and the method used was essentially an extension of the Roothaan open-shell SCF method with Lagrange multipliers.l9 The restriction to a few configurations was the consequence of an over-complicated method rather than a technical limitation of the computers available. In the early 1970s Levy3 showed that the use of an exponential form of the unitary transformation in connection with the Newton-Raphson formalism simplified the MCSCF formalism drastically. The general formulation of the MCSCF method is today essentially the same as outlined in ref. (3). One line of the modern development of the CI method essentially followed the 97 FAR CURRENT STATUS OF THE MC-CI METHOD general strategy outlined by Boys in the early 1950~.~ This method is known today as the conventional CI method and is almost unchanged.In this method the formulae for the Hamiltonian matrix elements are first constructed and stored. With these formulae the Hamiltonian matrix is set up and diagonalized by iterative techniques. The method is characterized by a high flexibility and a rather low upper limit for the allowed number of configurations of the order of lo4 terms. The most efficient variant of this method is probably the MRD-CI program by B~enker,~ where perturbation selection and perturbation extrapolation are two key features. The other line of evolution of CI methods started when the development of efficient general molecular Hartree-Fock programs was essentially completed in the early 1970s.A natural zeroth-order starting point was then a single Hartree-Fock determinant. To account for the major part of the correlation energy a wavefunction composed of single and double excitations from the Hartree-Fock determinant a HF-SD wavefunction was considered. That single and double excitations are the most important excitations for the dynamical correlation of the motion of the electrons follows directly from the two-particle nature of the electron-repulsion operator. Two efficient methods emerged very early in the solution of the CI problem for an HF-SD wavefunction. The PNO-CI method of Meyer6 used the simple structure of the two-electron problem to reduce the CI expansion to only a few hundred non-orthogonal terms.In the direct CI method the HF-SD CI problem was solved without constructing the Hamiltonian matrix explicitly.' The PNO-CI method of Meyer had an additional feature which has turned out to be very important for achieving a high accuracy. The CEPA correction for unlinked clusters was introduced based on the CP-MET theory of Cizeks and the work of Sinan~glu.~ With the CEPA correction a link was introduced between the variational CI approaches and the MBPT approaches. During the mid 1970s the cluster methods were developed towards the full solution of the CP-MET equations which was first done by Taylor et aZ.l0 A still simpler approximation than the CEPA correction is Davidson's often used correction.11 The fact that cluster corrections usually improve the results is now generally accepted and most CI calculations made today include such corrections.The question of size consistency is therefore no longer a dividing line between CI and MBPT methods. Since the early 1970sand the first efficient HF-SD methods the further development of methods for the electron correlation problem has gone in two different directions. First CI methods have been developed with a more general MCSCF zeroth-order starting point. Secondly MBPT methods have been developed which go beyond third-order perturbation theory. The essential equivalence between the first iterations of CI and MBPT up to third order was realized early in the 1970s [see e.g.ref. (12)]. In the direct CI program second-and third-order Moller-Plesset perturbation-theory results were routinely generated in all CI calculations. Note that these types of calculations did not become popular among ordinary chemists until recently almost ten years later when essentially the same algorithms were implemented in the GAUSSIAN 80 package.13 The first HF-SD programs already contained most of the features available in MBPT programs today. Apart from the very recent development of multireference coupled-cluster methods the main new development of the MBPT method has been the inclusion of the full fourth-order contribution. In going beyond the HF-SD approximation the emphasis has been in CI language on the inclusion of the major effects of triple and quadruple excitations.This has been combined with a requirement of size consistency which is however also required approximately in most CI calculations today. The major advantage of the MBPT approach in going beyond the P. E. M. SIEGBAHN 99 HF-SD approximation is that it is a well defined procedure and convenient to use. It is easy to predict that this is the procedure which will be of most use to chemists in high-accuracy applications. These calculations do not necessarily require expert quan tum chemists. The development of CI methods during the 1970s focused on an improvement of the zeroth-order wavefunction. The most important configurations usually described in terms of MCSCF orbitals define the reference space for single and double excitations.Most of the technical problems involved in the application of this procedure the MC-CI method were solved with the generalization of the direct CI method.14 The major disadvantage with this approach is that MCSCF is not always an easily defined procedure and can be very difficult to apply correctly by the user. This is particularly true when individual configurations should be selected. When orbital spaces should be selected as in the CASSCF method,15 the method can be made into a routine procedure if large enough CI expansions can be handled. When this is not the case CASSCF calculations still require expert quantum chemists. Note that if the MCSCF step is left out the multi-reference CI method can be made into a routine meth~d.~ One might well ask if it is necessary or desirable to use the MCSCF method.This question and others connected with applications of the MCSCF method are addressed in the next section of this paper. In section 2 it is argued that it is desirable in many cases to be able to make very large MCSCF calculations. The practical problem concerning this is found in the CI section where no efficient method exists today capable of making large general CI expansions. The only type of very long CI expansion which can be treated is the multi-reference SD expansion. However this type of wavefunction goes against the general idea of the MC-CI method in which all configurations in the zeroth-order space should be treated equally.The bottleneck to large general CI expansions is the large number of formulae needed on peripheral storage. This is the same problem that -prevented long HF-SD expansions before the direct CI method and should be solved by similar techniques. A method of treating long CASSCF expansions is discussed in section 3. Some applications of the MC-CI method to molecules containing transition metals are discussed in section 4. The emphasis is on the problems encountered in the applications and possible future developments suggested by these problems. It is predicted that the future development of the MC-CI method will be concerned with reducing the storage requirements for formulae still further. This requires a restructuring of the internal-space treatment.Some of the ideas used in the new method for large CASSCF expansions described in section 3 may be useful in this context. 2. COMMENTS ON MCSCF APPLICATIONS In this section some of the advantages and disadvantages of the MCSCF method as it is used today are discussed. Results from calculations on molecules containing second- and third-row atoms will be used to illustrate the points being made. The first question is how MCSCF should be used. This is a matter of choice. One way the MCSCF method has been successfully used is by trying to select only a few important configurations as in the OVC method of Wahl and Das for example.2 This procedure requires the user to have considerable experience and was developed at a time when only a few configurations could be handled technically.Today it is easily possible to select larger classes of configurations and this should almost always be an advantage. This type of method will therefore probably disappear just as methods which select only certain diagrams in MBPT have disappeared. The other extreme use 4-2 CURRENT STATUS OF THE MC-CI METHOD of MCSCF is the selection of all configurations in a certain orbital space as in the original CASSCF method.15 In this procedure the user should select orbitals rather than configurations which is of course much easier. Two well defined procedures can be set up for the orbital choice. The first procedure is the full-valence MCSCF where all configurations in the valence space are selected. The second preferable procedure is to select one weakly occupied orbital for each strongly occupied orbital.For H,O this leads to the addition of two orbitals outside the valence space and provides much improved results compared with the full-valence MCSCF procedure.16 The main problem with these simple well defined procedures is that the number of configurations increases rapidly with the number of valence electrons and is often unmanageable. One way to reduce the number of configurations without destroying the simple structure of the approach is to fix the number of electrons in each symmetry. In a 12-electron valence CASSCF on Cr this procedure leads to a reduction in the number of configurations by an order of magnitude without significantly affecting the accuracy. Even with this reduction in the number of configurations these well defined MCSCF approaches still lead to unmanageable CI expansions in many cases.Further reductions in the orbital space or configuration space will often require much experience of the user and is why the future of the MCSCF method as a standard tool for chemists depends very much on the possibility of being able to treat longer CI expansions. A method of treating longer expansions is described in section 3. The second question is when MCSCF should be used. The first area of application is trivially the cases where the single-configuration SCF method gives bad results. Such situations are rare for ground-state molecules containing only first-row atoms. For molecules with second-row atoms SCF results are often poor and with atoms of the third row it is more common for SCF to give bad results than reasonable ones.” For a small stable molecule like NiCO the binding energy is negative by 59 kcal mol-1 when the correct value should be 30 kcal mol-l.The wrong ground state is also predicted. For Fe(CO) and Fe(C,H,) the iron-ligand bond distance is incorrect by ca. 0.5 a.u. even though the bond (particularly for ferrocene) is very strong. These large errors are essentially corrected with a rather small CASSCF calculation where the active space is selected with the most preferable method described above i.e. one weakly occupied orbital for each strongly occupied orbital. The iron d orbitals are in this case either weakly or strongly occupied which leads to 10 electrons in 10 orbitals.It was also our experience in the case of Fe(CO) that it was nearly impossible to define a multi-reference CI based on the SCF orbitals and obtain a reasonable bond distance.18 For the seemingly simple diatomic molecules ClF and CCl in their ground states the SCF approximation also gives rather poor results. In these cases the bond distances are nearly 0.1 a.u. too short at the SCF level. A large MCSCF including the chlorine 3dshell was needed to obtain high-accuracy resultslg (see the discussion below). Again to extend this treatment to larger systems methods of handling large CI expansions in the MCSCF method are needed. The calculations described in the preceding paragraph have been examples of cases where an initial MCSCF is needed even when a multi-reference CI calculation is performed afterwards.There are other situations where an MCSCF calculation is needed and where a subsequent MC-CI calculation is not necessary. For molecular properties depending critically on the charge distribution such as multipole moments and transition moments a large MCSCF is sometimes required and is also sufficient.20 A CASSCF-type wavefunction also has invariance properties which greatly simplify the evaluation of the non-orthogonal transition moment. Another area where MCSCF will be very important in the future mainly for technical reasons is in the optimization of geometries of stable molecules and of transition states for reactions. Since the P. E. M. SIEGBAHN explicit derivatives with respect to geometrical parameters are much easier to calculate for wavefunctions where all the parameters of the wavefunction are variationally optimized the MCSCF method has a big advantage compared with CI methods.It is also our experience that a well defined MCSCF of the type discussed above will give accurate geometries. This is true even for transition-metal compounds with Cr as a notable exception. If saddle points for reactions involving transition metals are to be located with derivative methods very large CI expansions will again be required since the wavefunction commonly has quite a different form on one side of the barrier than on the other. Wavefunctions which go smoothly over the barrier are required to give meaningful derivatives.The final and in our opinion also the most important reason to carry out MCSCF calculations concerns the understanding of a given problem. Whenever an understanding is the issue the viewpoints will by necessity be very personal. Let us illustrate the arguments with the following calculations on CC1. The SCF and MC-CI calculations have been made.19 The other results from an HF-SD calculation are hypothetical and extrapolated from similar two-configuration reference CI calculations on CC1. The bond distance in CCl is too short at the SCF level by ca. 0.1 a.u. An HF-SD calculation using the SCF orbitals would as usual make the bond distance longer but probably still too short. A cluster correction of for example the CEPA type would then normally lead to an additional increase in the bond distance and it would not be surprising if the final result were close to the experimental value.Except possibly for the coefficient of the bond-dissociation configuration all coefficients in the wavefunction will be small of the order of 0.05 or less. Coefficients of this size appear for all molecules. The correct bond distance would be a valuable result and would show the capability of this simple well defined method. Let us now move to MCSCF calculations which have actually been performed. First the simple two- configuration MCSCF calculation which allows for proper dissociation gives a bond distance equally far from the experimental bond distance as the SCF result 0.1 a.u. but this time too long. This is a common effect.A full-valence CASSCF calculation including only the s and p shells gives the same poor result. More surprising is the fact that a multi-reference MC-CI calculation using the most important reference states does not significantly change this result. However including also the 3d shell of chlorine in the CASCF calculation has a dramatic effect on the bond distance and on the potential curve. Suddenly all spectroscopic constants come out in good agreement with experiment. Excellent agreement with experiment is finally obtained after an MC-CI calculation based on these latter CASSCF orbitals. Note that the 3d shell of chlorine is also very important to obtain an accurate ionization potential and electron affinity of the chlorine atom. Essentially the same sequence of results as for CCl was also obtained for ClF.By far the most important result of these MC-CI calculations in our opinion is the qualitative and general understanding of the effect of the chlorine 3d shell. This effect is not seen at all in an HF-SD +cluster-correction calculation which may otherwise show the same type of accuracy for all measurable properties. A few comments on the technical aspects of the MCSCF method may also be of interest. Powerful techniques exist today for converging MCSCF calculations [see for example ref. (21) and references therein]. Most of these quadratically convergent procedures have been implemented in our program. It may therefore be surprising that in practically all the applications we perform the simple first-order approximate super-CI method15 is the method which is used.The reason is that this method is in general much faster than the more elaborate methods particularly for large basis sets. A calculation on NO is illustrative. The first-order method required < 5 min per CURRENT STATUS OF THE MC-CI METHOD iteration and converged in 7 iterations. A second-order method took > 1 h for every iteration and was therefore not taken to convergence. Practically all of this time is taken up by the integral transformation. In our applications we have therefore only used second-order methods when the first-order method does not converge at all or for cases with few electrons. For cases where a high degree of convergence in the energy is required as for example when certain one-electron properties are calculated second-order methods may also be needed.Summarizing this section the MCSCF method is mainly useful for the insight it gives into the chemistry of problems and also for certain technical advantages which are achieved from invariance properties (transition moments) and the variationally optimized orbital parameters (geometry optimizations). It is further almost necessary to use an MCSCF starting point for multi-reference CI calculations whenever the SCF approximation gives bad results as is often the case for transition-metal compounds. In the systematic improvement of a multi-reference wavefunction where the SCF approximation still works reasonably well as for molecules containing first-row atoms MCSCF can also be of importance.However this is more debatable and depends on a given case. A straightforward and systematic application of the multi-reference CI methods based on SCF orbitals can be more effi~ient.~ Finally a systematic and well defined MCSCF approach which can generally be used for a large variety of problems will require the possibility of treating long CI expansions. A development in this direction is described in the next section. 3. A NEW CASSCF CI METHOD The direct CI method described in this section should be used as part of an MCSCF program and the goal is to perform large CI expansions. For this to be possible a minimum number of formulae must be stored. A secondary goal is that the algorithms should vectorize well.The method must also be able to treat certain reductions of the CASSCF expansions such as a limitation on the number of electrons in each spatial symmetry. The description of the method will be brief and the reader is referred to ref. (22) for further details. In the CI problem the Hamiltonian matrix is diagonalized either by an iterative method or by perturbation theory. In both cases the main computational step in each iteration is the construction of the vector o as where p,v are labels for the chosen configuration basis and c is the wavefunction from the previous iteration. The main working equation in the direct CI method can then be written as where and are the direct CI coupling coefficients. These expressions follow from the usual form of the Hamiltonian operator in its second quantized form where the Epq are the generators of the unitary group.From the expressions for thesecoupling coefficients it is obvious that the two-electron coefficients can be written as products of one-electron coefficients. The exact P. E. M. SIEGBAHN expression is obtained by inserting the projection operator over the complete configuration space the resolution of the indentity In the present method we use the fact that in the product in eqn (2) the same intermediate index K appears in both one-electron coupling coefficients. Only the one-electron coefficients are calculated and stored and these coefficients are ordered after K. This means that when the two-electron coefficients are constructed in the update formula (1) they can in principle be formed directly from multiplications within the same group of one-electron coefficients which are ordered sequentially.These multiplications are however only done indirectly (see below). When eqn (2) is inserted into eqn (1) we obtain for the term involving the product of one-electron coupling coefficients Aop = f C X A% AF:(pg I rs) c,. v,pqrs K This term can then be written after reorganization as which has been written to emphasize the matrix structure with Drs,K= C AFXc,,. 1’ Eqn (3) in matrix notation is written as Aop = iTr(Ap.1.D). The corresponding contribution AP to the second-order density matrix P which is needed in the CASSCF method is written as AP = +D*D’.The three matrices in eqn (3) have different characters. The symmetry-blocked integral matrix I is completely dense i.e. there are normally no zero elements. The matrix AP on the other hand is very sparse. The matrix D is in a normal CASSCF calculation rather dense; ca. 50% of the elements are non-zero. The trace of a product of three matrices requires one matrix multiplication and a scalar product. The matrix product can be performed between any two of the three matrices. Since the matrices D and I are independent of the index p it is clearly an advantage to perform the product between these two matrices. Some properties of the D matrix are worth noting. First it is straightforward to store the coupling coefficients required to form one matrix element of D sequentially on the formula tape which makes the formation of the D matrix easy.Secondly when there are many different spin-coupling possibilities there will be many CI coefficients c which will go into the same matrix element of D. This summation over spin couplings will therefore drastically reduce the number of necessary multiplications required to form o. This is an advantage which is not present in conventional CI techniques where each matrix element is multiplied by only one CI coefficient during the iterations. CURRENT STATUS OF THE MC-CI METHOD A detailed timing analysis of the present method as formulated in eqn (3) and a comparison with the conventional approach as formulated in eqn (1) is quite difficult. As mentioned in the introduction the goal with the present method was to reduce the elapsed time and more important the storage requirements.Such a reduction is sometimes worth even a large increase in C.P.U. time. (Compare this for example with the philosophy in the integral program DISCO where the integrals are recomputed in each SCF iterati~n.,~) It may seem as if the conventional approach must have an advantage in terms of C.P.U. time compared with the present approach since the summation over K in eqn (2) is performed prior to the CI step. This sum which is over the different spin couplings can in normal CASSCF calculations often be quite long owing to the large number of configurations with many open shells. There are two arguments against this simplified way of reasoning.First the C.P.U. time spent in reading and unpacking a large number of two-electron coupling coefficients is certainly not negligible. Secondly in the present method a presummation over spin couplings before the multiplication with the integrals is in any case made when the matrix D is formed. Conclusive timing comparisons between the two methods have not been made yet since optimal versions of the two methods are not available on the same type of computer. Preliminary experience shows however that on a vector processor the present method is usually much faster. On scalar machines the present method is also faster in most cases. Exceptions are cases where the number of configurations is substantially reduced owing to symmetry and excitation levels between symmetries in which case the methods are about equally fast.The largest case tried so far was a 30000-configuration calculation on Ni(C2H4)2 where the two-electron part required 10 s per iteration on a Cray-1 computer. Half of this time went into reading the one-electron coupling coefficients. A goal for further improvements of the method is therefore to avoid storing these coefficients. Algorithms for constructing these coefficients directly in the required order is under development. With the use of prototype formulae this turns out to be straightforward. 4. MC-CI CALCULATIONS ON TRANSITION-METAL SYSTEMS To perform calculations of qualitative accuracy on molecules containing transition metals is difficult with present techniques and will probably remain so for quite some time.To illustrate some of the problems appearing in these types of calculations two examples on rather small systems will be discussed. One system is NiCO where the ground state has only recently been determined. For this system it was possible following a reasonably well defined approach to design calculations which were small but which still yielded results of qualitative accuracy. The other system is FeO, for which present techniques are used to their limitations. Before we go on to discuss these examples the basis-set problems should be mentioned. For molecules containing only a few atoms of the first row it is today possible to saturate the basis set. This usually means much larger basis sets than are usually used however.Even if one atom in the molecule is from the second row basis-set saturation can be achieved. For molecules containing atoms of the third row this is no longer possible and is why calculations on molecules containing transition metals are only aimed at qualitative accuracy. Errors of 0.03 a.u. in the bond distances and 25% errors in the binding energies are what can normally be expected. Linear NiCO which is the most stable form of the molecule has often been used as a model for a complex with a typical transition-metal-ligand bond. It can also be used to simulate successfully bonding of molecules in positions on top of transition- metal surfaces. A one-configuration SCF treatment leads (as mentioned in section 2) P. E. M. SIEGBAHN to very poor descriptions for some of the states including the ground lC+ state which will be considered here.The binding energy of this state is negative by 59 kcal mol-1 when it should be positive by at least 30 kcal mol-l. The 3A state is better described at the SCF level and is why this state has earlier been predicted to be the ground state. The occupied orbitals of the carbonyl ligand are nearly unchanged compared with the free CO molecule in NiCO. This fact greatly simplifies the MC-CI treatment of this system. A reasonable description at the CASSCF level will be obtained without any strongly occupied ligand orbitals in the active space. A straightforward appli- cation of the most preferrable method of selecting orbitals as described in section 2 with one weakly occupied orbital for every strongly occupied orbital leads to 10 electrons in 10 orbitals.There is no problem to perform this CASSCF calculation particularly when spatial symmetry is used. As it turns out the two da orbitals stay practically doubly occupied so the CASSCF calculation could in fact have been per- formed with only 6 active electrons in 6 active orbitals. This is the minimum active space which gives a reasonable description for this state. Fixing the number of electrons in each symmetry to the Hartree-Fock occupation also works well in this case which leads to a compact description of the wavefunction at the CASSCF level. In this wavefunction there are 7 configurations (C,,) with coefficients > 0.10. No other coefficient is as large as 0.05.A multi-reference CI calculation with 7 reference states and a reasonably large basis set leads to (1-2) x lo5 configurations and the calculation requires ca. 2 h on a VAX 780 mini-computer using the contracted CI method.24 This is consequently a case of an unproblematic but by usual standards still rather large calculation. A very common complication is otherwise that those d orbitals which are not correlated in the CASSCF calculation will become mixed with ligand orbitals. These orbitals have to be cleaned from ligand contamination by some localization procedure. In our group we have simply used stepwise two-by-two rotations between the orbitals which are mixed to maximize the d contribution into as few orbitals as possible. This procedure has worked very well and constitutes a standard finishing step of each CASSCF calculation.Without this step almost random numbers will often be generated in the following CI calculation. A CI calculation correlating the d orbitals as they come out of a standard SCF calculation is consequently often meaningless for transition-metal compounds. A calculation on FeO is much more complicated than one on NiCO. In FeO the conformer with the ligand in a side-on orientation is more stable than that in a linear end-on orientation. In the side-on orientation the catalytic properties of the transition metal are more noticeable and the 0 ligand bond has started to break. This means that bonding-to-antibonding excitations on 0 cannot be disregarded. Another even more severe difficulty in the calculation on FeO as compared with NiCO is that the mixing between atomic states is in general more complicated on iron than on nickel.For NiCO in the ground lC+ state the only atomic state present on the Ni atom is the lD state with occupation d9s. This is a state which is favourable for bonding and is also a rather low-lying state on the atom. The corresponding 3F state on the iron atom with occupation d7sis far above the lowest 5D state which has an occupation of d6s2.Large mixings between d7and d6 states will therefore occur and it is not even clear what the spin state will be for FeO,. NiO is therefore simpler to treat than FeO,. Ideally a CASSCF calculation on FeO would include the 3d and 4selectrons on iron and the valence electrons (except 2s) on oxygen as active electrons.A full-valence CASSCF calculation is in this case at the limit of what can presently be handled. With the recipe of one weakly occupied orbital for each strongly occupied this leads to a CASSCF CI with 16 electrons in 16 orbitals which is an unmanageably long list of configurations. Whenever this situation occurs i.e. that the desirable calculation is CURRENT STATUS OF THE MC-CI METHOD outside the limitations of the program the user has to make a large series of test calculations before a reasonable calculation can be set up. The larger the CASSCF that can be handled the easier it is for the user which is why a method such as that described in the preceding section is particularly useful for transition-metal systems.Once a large CASSCF calculation has been done it is usually easy to design smaller calculations which also give reasonable results. The CASSCF calculations on transition-metal systems must always be followed by MC-CI calculations with as many reference states as possible. This is not only because dynamical correlation effects are generally important for these systems but also because such a calculation constitutes a test of the accuracy of the preceding CASSCF calculation. If a configuration outside the reference space appears with a large coefficient the calculation should preferrably by redone. It is our experience that if this configuration has an orbital occupied which is outside the CASSCF orbital space it is not enough to redo the CI step with this additional reference state.The CASSCF step also has to be redone including this extra orbital. Even when the coefficients for configurations including this orbital are as small as 0.05 in the CI calculation the inclusion of this extra orbital in the CASSCF calculation can completely change the bonding situation. An example is from a calculation on the 3B state of FeO, where the inclusion of a certain weakly occupied orbital in the first symmetry was critical. When this orbital was not active in the CASSCF step iron had a 3d occupation of 7.0 and 4s occupation of 0.2. The MC-CI calculation did not change these occupations significantly and no configurations with this orbital occupied appeared with coefficients as large as 0.05.Redoing the CASSCF calculation with this orbital active gave a 3d occupation of 6.4 and a 4s occupation of 0.8. The added active orbital had an occupation of 0.3. This is consequently a case where the MC-CI calculation did not give a clear indication that something was wrong. This can happen but is not normally the case. The only unusual feature of the MC-CI calculation was that the sum of the occupations for the orbitals in the first symmetry outside the reference space was as high as 0.05. The reason that there was no large coefficient in the MC-CI calculation is of course that the important orbital in the larger CASSCF calculation was spread out over very many orbitals in the smaller CASSCF calculation. To some extent this will always happen and is why a CI or MBPT calculation is so much harder to interpret and understand than an MCSCF calculation.The risk is obvious that all the chemistry the CI calculation will provide is the calculated number. A technical problem is also of general interest in the MC-CI calculations on FeO,. The 3B state which is one of the candidates for the ground state has 13 con- figurations with coefficients > 0.09 and a very large number of configurations with coefficients > 0.05. This means that a proper MC-CI calculation will have a very large number of configurations. This is however not the main problem in performing this calculation. What is even more problematic is that the reference states have varying occupations in at least 8 orbitals. This means that the evaluation of the formulae for the internal space will be time-consuming and even worse will require a large storage space.This calculation is in fact on the limit of what can be done today with the present direct CI technique. This is consequently the part where further method development of the multi-reference CI method is most urgently needed. Ideally the formulae for the internal space should be calculated during the CI iterations and never be stored. Work in this direction will most certainly be done the next few years. The method described in section 3 can also be useful in this context. It can be used without modifications for the formulae for the all-internal integrals. What remains is to develop the method further for the formulae for the integrals with one external index which presently requires most of the time and storage space.P. E. M. SIEGBAHN 107 Summarizing this section on MC-CI calculations on transition-metal compounds it is clear that many systems of chemical interest can be treated following simple principles. NiCO is an example of such a molecule. For partly dissociating systems such as FeO the situation is more difficult. What is required is MCSCF methods capable of handling a large number of active orbitals and MC-CI methods capable of treating large reference spaces. Both of these bottlenecks have to be removed by calculating the necessary formulae as they are needed without storing them. CONCLUSIONS In this paper the present status of the MC-CI method has been described.Examples are given particularly for molecules containing transi tion-metal atoms to illustrate the capability and the problems with the method. It is shown that the MCSCF calculation is the step which is most problematic. Well defined orbital selection schemes in the CASSCF method often require large orbital spaces. Methods for treating large CI expansions in the CASSCF method are therefore urgently needed. One such method is under development and was described in section 3. For the multi-reference CI step the main problem is usually not the total length of the CI expansion. Method development is instead required in order to allow larger reference spaces. This means an improved treatment of the formulae for the internal space. Ideally these formulae should not be stored but rather recalculated in an efficient way as they are needed.It is probably possible to use the new CASSCF CI method also in this context. C. C. J. Roothaan Rev. Mod. Phys. 1951 23 69; 1960,32 179. A. C. Wahl and G. Das in Methods of Electronic Structure Theory ed. H. F. Schaefer (Plenum Press New York 1977) vol. 3 p. 51. B. Levy Thesis (CNRS no. A0 5271 Paris 1971). S. F. Boys Proc. R. Soc. London Ser. A 1950 200 542. R. J. Buenker in Quantum Chemistry and Molecular Physics into the 80s,ed. P. G. Burton (University of Wollongong Wollongong 1980) p. 1.5.1. W. Meyer J. Chem. Phys. 1973,58 1017. B. Roos Chem. Phys. Lett. 1972,15,153; B. 0.Roos and P. E. M. Siegbahn in Methods of Electronic Structure Theory ed. H. F. Schaefer (Plenum Press New York 1977) vol.3 p. 277. J. Cizek J. Chem. Phys. 1966 45 4256. 0.Sinanoglu J. Chem. Phys. 1962 36 706. I" P. R. Taylor G. B. Bacskay N. S. Hush and A. C. Hurley Chem. Phys. Lett. 1976 41,441. l1 E. R. Davidson in The World of Quantum Chemistry ed. R. Daudel and B. Pullman (Reidel Dordrecht 1974). l2 P. E. M. Siegbahn in Proc. S.R.C. Atlas Symposium No. 4 Quantum Chemistry -the State of the Art ed. V. R. Saunders and J. Brown (Atlas Computer Laboratory Chilton Didcot Oxfordshire 1975). l3 J. S. Binkley R. A. Whiteside R. Krishnan R. Seeger D. J. DeFrees H. B. Schlegel S. Topiol L. R. Kahn and J. A. Pople GAUSSIAN 80 Quantum Chemistry Program Exchange (Indiana University 1980). l4 P. E. M. Siegbahn J. Chem. Phys. 1980 74 1647.B. 0. Roos P. R. Taylor and P. E. M. Siegbahn Chem. Phys. 1980,48 157. l6 B. 0.Roos Int. J. Quantum Chem. Symp. 1980 14 175. M. Blamberg U. Brandemark L. Pettersson P. E. M. Siegbahn and M. Larsson in Molecular Properties Proceedings of the CCPl Study Weekend Cambridge 1983 ed. R. D. Amos and M. F. Guest. H. P. Luthi P. E. M. Siegbahn and J. Almlof J. Phys. Chem. in press. l9 L. G. M. Pettersson and P. E. M. Siegbahn to be published. 2o M. Larsson P. E. M. Siegbahn and H. Agren Astrophys. J. 1983 272 369. J. Olsen D. L. Yeager and P. Jorgensen Adv. Chem. Phys. 1984 54 1. 22 P. E. M. Siegbahn Chem. Phys. Lett. 1984 109 417. 23 J. Almlof K. Faegri and K. Korsell J. Compact. Chem. 1982 3 385. 24 P. E. M. Siegbahn Int. J. Quantum Chem. 1983 23 1869.

ISSN:0301-5696    

DOI:10.1039/FS9841900097

出版商:RSC    

年代:1984

数据来源: RSC

摘要:

Faraday Symp. Chem. SOC.,1984 19 109-123 Structure Stability and Energetics of the Neutral and Singly and Doubly Ionized First- and Second-row Hydrides BY SUSAN A. POPEAND IAN H. HILLIER* Chemistry Department University of Manchester. Manchester M 13 9PL AND MARTYN F. GUEST S.E.R.C. Daresbury Laboratory Warrington WA4 4AD Received 13th August 1984 Geometry-optimization studies have been carried out on the neutral and singly and doubly ionized states of the first- and second-row hydrides AH?+ (A = nitrogen oxygen phosphorus sulphur; m = 0 1,2; n = 1,2,3; except H,O and H,S) and AHF+ (A = nitrogen phosphorus; m = 0,1,2). Calculations have been performed at both the SCF and CASSCF levels. Comparison with experimental geometries show that for the first-row hydrides the underesti- mation of the bond lengths and overestimation of the bond angles found at the SCF level are successfully corrected at this MCSCF level.For the second-row hydrides the errors in the SCF values are smaller whilst the subsequent MCSCF corrections are close to those for the first- row hydrides yielding poorer agreement with experimental geometries. Barriers to deprotonation of the dications have been calculated to interpret mass-spectrometric charge-stripping data. All the dications studied except OH2+,NH2+,H202+and NH:+ have relatively large barriers in qualitative agreement with experiment. The calculated adiabatic ionization energies for the monocations of nitrogen and oxygen are except for NH significantly larger than the experimental values.This discrepancy is unlikely to reflect inadequacies in the calculations and points to excited vibrational states of the monocations being involved experimentally. One of the most successful examples of electronic structure theory complementing experiment is in the interpretation and understanding of molecular ionization phen0mena.l The use of ab initio molecular-orbital (MO) calculations with or without the inclusion of correlation effects to calculate the vertical molecular ionization energies obtained from photoelectron spectroscopy is well In such work the calculation of the differential correlation energy between parent molecule and ion may present a severe problem since quantitative estimates demand that a large percentage of the correlation energy is recovered for both species.The more demanding computation of adiabatic ionization energies usually requires the theoretical determination of the equilibrium geometry of at least the molecular ion and has benefitted from the comparatively recent development of analytic gradient techniques.* Such methods allow the determination of not only energy minima but also transition-state structures so that the stability as well as the equilibrium structures of the ionic species may be studied.j Although single ionization energies can be measured by a number of techniques double ionization energies or appearance energies of doubly charged ions are more difficult to obtain experimentally. Recent developments in mass spectrometry by Beynon and c~workers~-~ have yielded new information on both the adiabatic 109 STRUCTURE OF HYDRIDES ionization energy of the cation and the stability of the resulting dication for a series of simple hydrides.It is the purpose of the present paper to provide a theoretical interpretation of these data and to present such results in the context of a more general study of the structure stability and energetics of the neutral and singly and doubly ionized first- and second-row hydrides. EXPERIMENTAL BACKGROUND The charge-stripping method6 is now an established technique for measuring the difference between single- and double-ionization energies. When ions (M+) having high translational energy collide with neutral gas molecules (N) in a mass-spectrometer drift region the charge-stripping process M++N -+ M2++N+e (1) can occur.The minimum loss of translational energy (Qmin)of M+ necessary to allow observation of M2+ may be equated to the first adiabatic ionization energy of M+. Such experimental studies* on the hydrides derived from NH, H20 and H2S imply that all species except H202+,OH2+ NH2+ and NHg+ have lifetimes of several microseconds suggesting a significant barrier to the strongly exothermic process AH:+ +AH:- +H+. (2) In this paper we report calculations of the equilibrium structures of the species AHF+ (A = nitrogen oxygen phosphorus sulphur; nz = 0 1,2; n = 1,2,3; except H30 and H3S) and AHF+ (A = nitrogen phosphorus; m = 0,1,2) thus allowing predictions of the adiabatic ionization energies of the neutral and cationic species to be made.A number of geometry-optimization studies particularly of the neutral and singly charged species involving first-row atoms have been reported by other workers. We include our results on these species to allow for a consistent comparison between all 40 species studied herein. We also report calculations of the structure and energetics of the transition state associated with the deprotonation reaction (2) involving all 14 dications mentioned above. COMPUTATIONAL DETAILS The following gaussian basis sets were employed throughout this work (i) The split-valence 3-21G and 4-31G basis sets due to Pople et aZ.l0(ii) A triple-zeta (TZ) valence basis for hydrogen (5s/3s);11 for first-row atoms (10~6p/Ss3p);~l for second- row atoms (12S9p/6s5p).12 The TZ basis was augmented with a Rydberg s gaussian on the heavy atom ([ = 0.02) to permit a description of those radical species deemed to exhibit a ground electronic state of Rydberg character such as NH4.13 (iii) Polarization was introduced by augmenting the TZ basis by appropriate functions on each atom.In this basis (denoted TZP) a hydrogen p function with exponent 1.O was used while for the heavy atoms sets of six d functions having the following exponents were employed nitrogen 0.98; phosphorous 0.465; oxygen 1.28 ;sulphur 0.542.14 (iv) A more extensive polarization basis denoted TZ +2dlp comprising the TZ basis augmented by two sets of d-polarization functions on each heavy atom15* l6and a single set of 2p functions on hydrogen.14 Two levels of theory have been employed in obtainingminima on the potential-energy surfaces and in locating any barrier associated with the deprotonation reaction (2).(a) Spin-restricted Hartree-Fock SCF-gradient calculations. Singlet states of the various diradical species involved (e.g. H2S2+ PH; NH; H202+) were represented S. A. POPE I. H. HILLIER AND M. F. GUEST by two-configuration SCF wavefunctions to ensure consistency with the use of RHF methods for the corresponding triplet states.17 These calculations were performed within the GVB- 1/PP formalism.lS (b) MCSCF-gradient calculations using the complete active space SCF (CASSCF) method.lg? 2o In the present calculations the 1s orbital of the first-row atoms and Is 2s and 2p orbitals of the second-row species were chosen to be inactive i.e.doubly occupied in all configurations. The active space was generally taken to be a full-valence-orbital space with the CASSCF geometric structures thus obtained subsequently investigated by extending this space. The CASSCF calculations were performed using the TZP and TZ+2dlp basis sets. Subsequent to optimization of the HF/TZP CASSCF/TZP and CASSCF/TZ + 2dlp structures single calculations were conducted at these points with highly correlated wavefunctions. Single-reference state and multi-reference state CI cal- culations (hereafter referred to as SDCI and MRDCI respectively) were performed including all single and double excitations of the valence electrons using the direct-CI method,21 utilizing in general the appropriate set of molecular orbitals derived in obtaining the optimized geometry.Two series of MRDCI calculations were under- taken. (i) All configurations having coefficients > 0.05 were included in the reference set. (ii) More extensive calculations on the triatomic hydrides were performed in which the reference set was taken to comprise all configurations which could be generated by single and double excitations of the valence electrons amongst the active orbitals of the CASSCF zeroth-order wavefunction with respect to the leading term of that wavefunction (usually the HF configuration). Generating all single and double excitations from such a reference set which typically comprised some 100 terms led to CI expansions of up to 150000 configurations.It is hoped that the major part of the dynamic correlation effects would be recovered from such a treatment,, hereafter referred to as MRDCI2. We adopt the notation MRDCI2/TZ +2dlp//CASSCF/TZP for example to indicate a multi-reference CI calculation with the TZ +2dlp basis set at a geometry optimized at the CASSCF level with the TZP basis. Finally harmonic vibrational frequencies were obtained for stationary points on the HF/3-2 1G surface allowing for the determination of zero-point vibrational energies (ZPVE). These energies were scaled by 0.9 when used in the evaluation of reaction energies and barrier COMPUTATIONAL RESULTS PREDICTED GEOMETRIES AND ELECTRONIC STRUCTURES AH MOLECULES AND IONS We consider here the neutral and mono- and di-cation structures for AH molecules where A is nitrogen oxygen phosphorus or sulphur.Omitting the inner-shell molecular orbitals the electronic structures of these species may be specified in terms of the occupancy of the (lal lh, 2a1 lb,) MO for C, symmetry and the corresponding (1 ag 1a, 1nu)MO in Dmh.The states considered here involve double occupancy of the lal(lag) and lb (la,) MO. It is well known that the equilibrium bond angle of these molecules may be discussed by the use of Walsh’s rules,24 focusing on the 2a1 MO which favours a bent rather than a linear structure and leading to the configurations and approximate geometries shown in table 1. The 8-valence-electron systems H,O and H,S. The optimum geometries from SCF and CASSCF calculations employing the 4-3 I G TZ TZP and TZ +2dlp basis sets on H,O and H,S are given in table 2 along with the results of previous studies STRUCTURE OF HYDRIDES Table 1.Low-lying states of AH, AH and AH:+ ~~ valence-electron configuration approximate molecule Dmil c2 bond angle/" HZO H2S 10; 10;1.yq laf lbi2a lb?(lA,) 90 H,O+ H2S+ PH, NH 10; 10; l.",("rI,) laf lb2,2af lb;(2Bl) 90 la? lbi 24 lb?(2A1) 120 'qlAg 1.; 10;10; ("c,, H202+,H2S2+ NH PH; 1a; I b; 24 1b;(3~,) 120 la 16 2a lbY(lAl) 90 laf lb 24 1b?(lAl) 180 NHi+,PHi+ 1.; 10; 1.',(2nn,) la lbi 24 lbyA,) 120 la? lb 2ay lbi(2Bl) 180 Table 2. Optimized geometrical parametersa of H,O and H,S treatment basis r(H-0) LHOH r(H-S) LHSH HF 4-3 1G 0.950 111.2 1.354 95.5 HF DZ25 0.951 112.5 HF TZ 0.950 112.6 1.362 96.0 HF DZdp25 0.944 106.7 HF TZP 0.941 107.1 1.333 94.4 SDCI DZdpZ5 0.960 105.0 CASSCF/32 10 TZP 0.964 103.7 1.359 93.2 CASSCF/3210 TZ2d2~~~ 0.965 103.0 CASSCF/4220 TZP 0.961 105.9 1.348 93.0 CASSCF/4220 TZ +2dlp 0.962 105.1 1.333 92.2 CASSCF /4220 TZ2d2pZ6 0.964 104.8 UMP4(SDQ) 6-3 1G**27 0.957 102.7 UMP2 6-3 lG*28 0.968 103.9 ex tended29 0.955 105.4 experiment3O,31 -0.957 104.5 1.328 92.2 a In all tables distances are in ingstroms (A) and angles in degrees (").including the CASSCF study on H,O by Roos.,~The general trends in predicted geometries with improved basis are well illustrated by these molecules. Considering the water molecule the bond lengths are predicted quite well at the HF/4-31G level although the bond angle is too large by some 7".No improvement in the geometry is found upon extending the s,p basis while the inclusion of polarization functions reduces the bond angle by 4.1" which is accompanied by a marked shortening of the bond length. The CI/DZdp calc~lation~~ gives good agreement with experiment by increasing the bond length and decreasing the angle. The dominant valence-electron configuration for the 8-valence-electron systems (table 1) is in C, symmetry (1~1,)~(2a,)~( 1bJ2(lb,), the unoccupied valence orbitals being 3a and 2b,. Geometry optimization using the CASSCF method26 suggests that employing this full-valence orbital space (with an associated very short CI expansion of 37 configurations) leads to rather disappointing results the bond angle of 103.0" S.A. POPE I. H. HILLIER AND M. F. GUEST underestimating the experimental value by 1.5". The relative 'failure' of this calculation denoted CASSCF/3210 (the notation describing the number of orbitals in the four symmetry species a, b, b and a, respectively) has been rationalized26 in the uneven way the four electron pairs are correlated by the two orbitals 3a and 2b,. These orbitals are used to correlate two of the strongly occupied orbitals one lone-pair orbital (2a,) and one OH bonding orbital (1 b,). Clearly a balanced treatment requires all four electron pairs to be correlated to the same extent. In a CASSCF/4220 treatment (492 configurations) the 4a orbital is used to correlate the la (OH bonding) and the 2b1 correlates the 7c lone-pair orbital lb, such a treatment leads to much better agreement with experiment.26 The present results suggest that the CASSCF optimum geometries are sensitive to basis-set variations thus at the CASSCF/4220 level use of the TZP and TZ+2dlp basis set leads to a small overestimation of the bond angle compared to the (6s4p2d/5s2p) basis of Roos with values of 105.9 and 105.lo respectively compared to the estimate of 104.8' from ref. (26). The general pattern in predicted geometries of the FA ground state of H,S as a function of theoretical treatment is similar to that in H,O with configuration interaction shifting the optimum geometry to longer bond lengths and smaller bond angles.The agreement between the experimental S-H bond length and the HF/TZP estimate is perhaps surprising [see also ref. (33)] while the marked decrease in this quantity on improving the CASSCF treatment is worthy of note. Thus the CASSCF/3210 estimate of 1.359 A decreases by 0.01 1 A on extending the active space and by a further 0.015 A on improving the basis set. Thus the final CASSCF/4220 estimate of 1.333 A and a bond angle of 92.2" is in surprisingly good agreement with the experimental geometry of 1.328 A and 92.2°.30 This agreement should be viewed with some caution the natural orbital occupation numbers reveal that no effective correlation of the la orbital is achieved even at the CASSCF/4220 level. In the latter calculation the 4a orbital appears to act as a second correlator for the 2a valence orbital rather than correlating the 1a orbital.The 7-valence-electron systems NH, OH PH and SH:. The ground state (X2Bl)of these species corresponds to a half-filled 1 b MO and a doubly occupied 2a1 MO lead-ing to an expected bond angle far from linearity whilst the first excited state (A,A,) has this occupancy reversed and an expected larger bond angle. Our optimized geometries (table 3) are in accord with these predictions and with previous calc~lations.~~~ 399 We note that at the HF/TZP level bond lengths in the X2B,states of NH and H,O+ are underestimated by 0.015 and 0.019 A respectively whilst the bond angles are overestimated by 1.9 and 2.2' res ectively. Both effects are compensated for at the CASSCF level increases of 0.025 x(H,O+) and 0.029 A (NH,) in bond lengths and decreases of 3.9" (H,O+) and 3.1" (NH,) in bond angles are found relative to the SCF estimates in the CASSCF/3210 calculations.A more extensive MCSCF treatment at the CASSCF/4220 level with a TZ +2dlp basis yields bond lengths within 0.01 A and bond angles within 1" of the experimental data. The situation is less satisfactory for the second-row hydrides. Although the bond lengths are again given to within 0.01 A by this CASSCF treatment the overestimation of the bond angle given in the HF/TZP calculation is not sufficiently compensated being too large by 1.8" and 1.5" for H,S+ and PH, respectively. Similar effects are evident in our treatment of the *A,excited states (table 3); the good agreement found between the CASSCF geometry and experiment in NH (and for H,S+) is not evident for PH,.The 6-valence-electron systems NH PH H,02+ and H,S2+.By analogy with the well known situation in the isoelectronic CH molecule,41 candidates for the ground states of these molecules are 3B,or ,A (in C2J and 31=; or ,As (in Dmh)(table 1). STRUCTURE OF HYDRIDES Table 3. Optimized geometrical parametersa for AH molecule HF/TZP CASSCF/TZPb experiment 0.980 112.7 1.005 108.8 0.999 1 10.534 1.009 105.2 1.348 95.4 1.038 102.1 1.373 94.1 1.024 103.331p 35 1.358 92.536 1.414 93.7 1.442 92.9 1.428 91.531737 0.976 180.0 0.987 180.0 - 0.985 142.9 1.351 126.7 1.000 143.1 1.370 126.3 1.004 lU3l I .369 1 2736338 1.387 122.0 1.413 121.5 1.403 123.131337 1.019 143.1 1.037 149.7 1.173 180.0 1.182 180.0 1.397 121.0 1.421 122.0 1.428 128.6 1.447 131.8 1.031 110.0 1.058 106.2 1.412 94.4 1.439 94.1 1.402 97.7 1.428 96.6 1.145 180.0 1.169 180.0 1.464 180.0 1.496 180.0 1.449 125.2 1,483 124.8 a The A-H bond length is given first followed by the HAH bond angle.CASSCF/3210. we In agreement with previous MRDCI cal~dations~~ find the ground state of NH; to be the quasi-linear 3B1. The 1A,-3B splitting at the (MRDCI2/TZ + 2dlp//CASSCF/TZ + 2dlp) level 3 1.2 kcal mol-l,* is close to the MRDCI value 42 (29.9 kcal mol-l). In contrast the stabilization of the 2a1 MO in PHZ results in a lA1 ground state for this molecule predicted at the CI/TZP//HF/TZP level to lie some 15 kcal mo1-1 below the 3B state.A similar situation is evident in the dications of H20 and H2S.The vertical 2Bl-2Alenergy separation is ca. 1 eV larger in H2S+ than in H20+ so that ionization to the ground state of H2S2+is from the lb MO leading to a strongly bent lA1 ground-state dication with an estimated (at the zero point corrected CI/TZP//HF/TZP level) 1A1-3B1separation of 2.5 kcal mol-l. In contrast ionization of H20+(2B,) is from the 2a1 MO leading to a linear 3E; ground state of H202+(table 3). The 5-ualence-electron systems NHi+ and PHi+. Ionization from the quasilinear 3B1 ground state of NHZ leads to a linear dication (X2n,) with the valence configuration 10; la In,. In contrast stabilization of the 2a1 MO of PH; leads to a bent 2Al ground state of PHi+ arising from ionization from this MO.The strongly P-H bonding character of the 2a1 MO leads to an increase in the bond angle from 94.1 to 124.8' upon ionization. The first excited state of PHi+ which is found to be linear (",) is predicted to lie 19.4 kcal mol-l above the ground state at the CASSCF/TZP//CASSCF/TZP level. AH MOLECULES AND IONS The possible low-lying states of the AH molecules and ions considered here are given in table 4 and the results of bond-length optimization calculations are shown in table 5. These species have been studied extensively by Meyer and Ros~us.~~~~~ The * 1 cal = 4.184 J. S. A. POPE I. H. HILLIER AND M. F. GUEST Table 4. Low-lying states of AH AH+ and AH2+ electronic molecule configuration state OH SH 1 a22a2 1n3 2II OH+,SH+ PH NH 1 a22a2 1 712 3x-OH2+,SH2+,PH+,NH+ la228 1711 2n 4.2x-2A 2z+ 1022a’ 1 n2 9 PH2+,NH2+ la22a2 lx+ 3,1n 10220171 Table 5.Equilibrium bond lengths of the ground states of AHm+ ~~ molecule ~~ HF/TZP CASSCF/TZP experiment43,44 1.020 1.049 1.037 1.049 1.078 1.070 - - - 0.952 0.976 0.971 1.005 1.029 1.029 - - - 1.419 1.443 1.422 1.417 1.448 1.425 1.452 1.474 - 1.337 1.363 1.341 1.354 1.374 1.364 1.425 1.451 a Repulsive potential curve. CASSCF results utilized the full valence space la,2a,3aand ln MO. The ground-state symmetries of the six- and seven-valence-electron systems present no problems being 3E-and 211 respectively. However for the five-valence-electron systems 211 and 4Z-are the contenders for the ground state.For NH+ and PH+a 211ground state is found the X 211-A 4C-separation being greater for PH+ (1.6 eV45)than for NH+ (0.07 eV49. We find a similar pattern in the isovalent OH2+ and SH2+ radicals. Thus SH2+ exhibits a 211ground state whilst calculations on OH2+ at the geometry of OH+ suggests that the 4Ec-state lies some 1.6 eV lower in energy than the 211state. However no minimum is found in the Hartree-Fock potential curve of either state with any of the basis sets. Clearly the ground-state potential curve is repulsive suggesting that dissociation to O+(4S)and H+ will be instantaneous. The ground state of PH2+ is calculated to be lZ+(la22a2). However for NH2+ calculations at the NH+ bond length suggest a 311ground state.As for OH2+ this Hartree-Fock state is found to be repulsive for all basis sets used so that we again conclude that dissociation of NH2+ to N+ and H+ would be spontaneous. Comparison with the experimental bond lengths shows that for the first-row species the CASSCF calculation corrects the inadequacies of the HF/TZP calculations. However for the second-row species the CASSCF bond lengths are uniformly too long. For SH we find that increasing the CASSCF space from (30 In) to (40 2n) whilst increasing the correlation energy recovered from 0.019 to 0.059 a.u. changes STRUCTURE OF HYDRIDES Table 6. Ground states of AH, AH and AH$+ D:m c,tj ____~___ molecule configuration state configuration state Table 7.Optimized geometrical parameters for the AH speciesa molecule HF/TZP CASSCF/TZP experiment 0.998 108.9 1.010 120.0 1.022 106.0 1.030 120.0 1.013 1O7.Og7 - 1.084 120.0 1.110 120.0 I 0.957 119.9 -. - 1.055 120.0 1.078 120.0 - 1.410 95.6 1.392 1 12.4 I .424. 120.0 1.439 94.3 1.416 113.0 1.451 120.0 1.412 93.448 - 1.344 96.8 1.389 116.0 1.410 118.2 - - The A-H bond length is given first followed by the HAH bond angle. the bond length by only 0.001 A. Both the (30 171) and (40 271) CASSCF bond lengths decrease by 0.007 A on utilizing the TZ +2dlp basis However the calculated bond length is still 0.014 A larger than experiment. AH MOLECULES AND IONS In table 6 we show the ground-state configuration of these species for D,,and C, symmetry.Table 7 lists the optimized geometries at the HF/TZP and CASSCF/TZP levels the full valence space of seven MO being used in the CASSCF calculations. The geometry of these species is in general determined by the occupancy of the 2a1 lone-pair MO (1 a; in D3h).49For those species with more than six valence electrons this MO is occupied and its increased stability away from a planar structure implies a strong trend towards a pyramidal equilibrium geometry for such species. It is well known that the eight-valence-electron molecules (NH, PH, H,O+) are pyramidal and many calculations have been carried out to predict their inversion Ionization from the 2a1 MO is less likely to lead to a planar structure for the second row than for the first row species. Thus NH$ is planar,52 whilst PH$ has been shown both e~perirnentally~~7 j6 to be non-planar.Large basis SCF-CI 54 and the~retically,~~? calculations on SiH, PH,+ and SH:+ predict all three to be n~n-planar.~~ The geometries shown in table 7 reveal the behaviour previously noted between first- and second-row species. Thus the underestimation of the experimental bond length at the HF/TZP level is far greater in the first-row species (NH, 0.015 A) than in the second-row species (PH, 0.002 A). The constant increase in redicted bond length at the CASSCF level ranging from 0.020 A (NH;) to 0.029 1(PH,) relative to the S. A. POPE I. H. HILLIER AND M. F. GUEST Table 8. Optimized geometrical parameters of AH, AH and AH:+ value molecule symmetry state parameter HF/TZP CASSCF/TZP r(N-H) r(N-H) 1.018 1.011 1.031 r(N-H) 1.125 r(N-H) r(N-H) 1.104 1.102 1.125 1.126 LHNH 147.2 146.2 r(P-H) r(P-H) r(P-H) r(P-H) 1.414 1.389 1.495 1.481 1.415 LHPH 135.1 r(P-H) 1.882 1.877 r(P-H) 1.416 1.438 LHPH 95.6 95.5 HF/TZP estimates results in poorer agreement with experiment for PH at the MCSCF than at the SCF level.The decrease in bond angle found in both NH and PH at the correlated level of treatment leads however to improved agreement with experiment. AH MOLECULES AND IONS The optimized geometrical parameters for the AH species are summarized in table 8 the CASSCF studies utilizing the full valence space of eight MO. In agreement with previous work,13. 57 NH (G)corresponds to a minimum on the potential-energy surface lying 3.3 kcal mol-1 below H +NH at the CI/TZP//TZP level.Zero-point corrections decrease this value to -5.2 kcal mol-l again suggesting that the stability of this species is controlled by zero-point effects. Calculations on PH suggest that the tetrahedral structure is similarly a minimum on the ground-state potential-energy surface but in contrast to NH the dissociation of PH to PH +H is predicted to be exothermic at all levels of treatment being 28.5 kcal mol-l when zero-point corrections are applied to the CI/TZP//TZP estimate. Both NH and PH have tetrahedral structures our values of the N-H bond lengths of 1.01 1 A (HF/TZP) and 1.031 A (CASSCF/TZP) being close to previous SCF and CI values of 1.0107 and 1.0185 A re~pectively.~~ We find a similar increase in the P-H bond length of PH being 1.389 A at the HF/TZP level and 1.41 5 A at the CASSCF/TZP level.The dications NHi+ and PHi+ have seven valence electrons and as in the isovalent CH,f cation distortion from G is expected.5s Investigations of both tetrahedral and lower-symmetry structures of NHi+ reveal at least three stationary points of G,D2d and D,,symmetry (table 8). Preliminary studies suggest that C, structures converge to a Dadconfiguration whilst C, structures dissociate to NH; +H+. Lower-symmetry structures are unlikely since dissociation would probably occur owing to charge localization. Both SCF and CI calculations place the 2T state significantly higher in energy than the distorted Dzd and Ddhstructures.We find the 2A (D2J state to be a minimum on the potential-energy surface whilst 2A2u(D4J corresponds to a STRUCTURE OF HYDRIDES Table 9. Calculated and experimental adiabatic ionization potentials (eV) of the neutral hydrides ~~ HF CASSCF CI molecule state TZ TZP TZP TZ+2dlp TZP MRDCI2 experiment 2A 4.0 4.0 -4.4 -4.760,61 'A 8.5 8.7 8.7 -9.6 -10.262 2Bl 9.3 9.6 9.6 10.9 10.5 10.9 1 1.563 3Z-12.8 12.8 12.3 -13.1 -13.564 2Al -3.5 -3.8 IA1 8.7 8.6 8.7 -9.3 -10.054 2Bl 9.1 8.6 8.6 -9.4 -3C-9.8 9.6 9.3 -9.5 -9.865 2A' 4.6 4.6 -5.1 lA1 11.0 11.0 11.0 12.3 12.0 12.3 12.666 217 11.4 11.4 11.3 -12.3 -1 3.06' 'A 9.5 9.4 9.2 9.3 9.8 10.0 10.566 2l-I 9.3 9.2 9.1 -9.7 -10.46s transition structure with one imaginary frequency.Similarly for PH:+ we find the 2& (q)state to be significantly higher in energy than the CS2, and Dzd structures. The C3vstructure (,A,)corresponds to an energy minimum with one very long P-H bond. The Dzd (2A,) structure is not a minimum at the HF/3-21G level but a hill-top characterized by a doubly degenerate imaginary frequency. At the HF/TZP level this Dzd structure lies 23.8 kcal mol-l above the C, structure. ADIABATIC IONIZATION POTENTIALS THE NEUTRAL HYDRIDES In table 9 we show the calculated first adiabatic ionization potential of the neutral hydrides. As expected the SCF values are too small compared with experiment an increase towards the experimental value being evident as the basis size and level of correlation are increased.A balanced treatment of correlation in both molecule and ion must be capable of recovering 90-95% of the valence-shell correlation in order to yield ionization energies accurate to 0.1 eV. This percentage is clearly not obtained using the present TZP and limited MRDCI calculations so that the CI/TZP calcu- lations in table 9 underestimate the ionization energies by ca. 0.7 eV. Extensive calcu- lations which we have carried out of the adiabatic ionization energy of H,S illustrate the slow convergence of the calculated value. Thus using a (12~9p2dlf/7~5p2dlf) sulphur basis and a (5s2p/3s2p) hydrogen basis at the SDCI level with estimates of the importance of higher excitations being obtained using the modification of Davidson's c~rrection~~ due to Pople et aZ.,'O leads to a value of 10.22 eV compared with the experimental value of 10.48 eV.THE CATIONIC HYDRIDES The calculated and experimental8 ionization energies of the cationic hydrides are compared in table 10. We find the same trends with successive improvements in the theoretical treatment as was found for the neutral hydrides. Thus the HF/TZP estimates increase as more account is taken of the differential correlation between the S. A. POPE I. H. HILLIER AND M. F. GUEST 119 Table 10. Calculated and experimental adiabatic ionization potentials (eV) of the cationic hydrides HF/TZP CASSCF/TZP CI/TZP CI + cation state //TZP //CASSCF/TZP //HF/TZP ZPVE experimental*.a NH ,A 23.3 23.7 24.0 23.6 24.5 NH; 2Ai 22.5 22.3 23.2 23.0 22.2 NH; 3B 24.0 23.8 24.5 24.4 23.3 NH+b 25.6 -26.9 -25.0 PH lA 20.0 20.7 20.3 PH,f 2Al 17.6 17.4 18.0 18.0 PH; ,A 18.9 18.7 19.2 19.2 -PH+ 18.4 17.7 18.7 18.7 OH 'A 22.3 -23.2 22.9 22.5 OH 2Bl 23.5 23.6 24.5 24.3 23.5 OHtb 3C-28.4 28.5 29.6 -29.0 SHT 'A 19.2 -19.8 19.6 21.4 SH; 2BB 20.0 20.1 20.9 20.8 21.0 SH+ 3C-21.7 21.4 21.9 21.9 21.2 a Accuracy is k0.3 eV except for NH; NH+ OH and OH+ for which the accuracy is estimated to be f1.0 eV.Values derived at the monocation equilibrium geometries. mono- and di-cation. Little improvement at the CASSCF level is found for the ionization energies of both the neutral and cationic hydrides since this treatment favours the species with the fewer number of electrons. The most extensive calculation carried out that of the ionization energy of H2S+ which parallels that of H2S previously described yields a value of 2 1.12 eV 0.3 eV larger than the best calculated value in table 10.We note that differences in zero-point energy between the mono- and di-cation which act to decrease the calculated ionization energy are more important than for the neutral hydrides since the bonding in the parent monocation is significantly stronger than in the corresponding metastable dication. This effect is found to be greatest for NHZ and PHZ decreasing the ionization energy of both species by 0.4 eV. Comparison of the calculated adiabatic ionization energies of the monocations with the experimental charge stripping values (table 10) shows that for all the species NH; and OH; except NHZ the calculated values are 0.5 to 1.9 eV greater than the experimental values.A similar effect has been noted for the CHL 72 For the monocations SHL we find no systematic deviation between theory and experiment and for the PHL species no experimental data are available. The origin of these discrepancies is unclear. It is possible that transitions from vibrationally excited monocations to ground-state dications provide an explanation where the theoretical ionization energy is larger than the charge-stripping value. STABILITY OF THE DICATIONS The calculations previously described have shown that of the fourteen dications studied all except OH2+ and NH2+are characterized by a local minimum on the potential-energy surface. The stability of these cations will clearly depend upon the magnitude of the barrier for the deprotonation reaction (2).We estimate that the lowest asymptote is in each case AH:- +H+ while the lowest attractive channel generally corresponds to AH2,$_ +H. The most probable mechanism for reaction (2) is thus an initial motion along this attractive channel followed by an electron jump STRUCTURE OF HYDRIDES -396.85--397.001 -397.05 1 --397.10 I11III 12 3 4 5 6 7 8 9 10 r(S-H)/ A Fig. 1. Potential curve for SH2+at various levels of treatment. corresponding to dissociation on the former repulsive curve. This is clearly illustrated in the potential curves for the 211state of SH2+computed at three levels of treatment (fig. 1). Whilst the HF/TZP curve is expected to provide a good zero-order description of the reaction the close agreement in both the height and location of the barrier for the three calculations is gratifying.The calculated deprotonation energies and activation barriers are given in table 11. The major trend observed is for an increase in the barrier between the first- and second-row species. Note however that the ground state of the monocations is not obtained in all cases. Thus deprotonation of SH;+(lA,) and NH;+(lA;) leads to the excited states SH+(lA) and NH;(lA,). We see that correlation generally has an effect on the deprotonation energies not exceeding 9 kcal mol-l although no consistent trend is apparent. The zero-point corrections always lead to a larger deprotonation energy owing to the additional degrees of vibrational freedom in the dication.Activation barriers of < 10 kcal mol-1 are found for deprotonation of H202+and NHt+. The detection of these dications together with OH2+ and NH2+ which are predicted to have repulsive potential curves would thus not be expected in the charge-stripping experiments. Deprotonation of the remaining dications is ac-companied by a substantial activation barrier indicating that these species should be observed. These predictions are in complete agreement with experiment.8 The transition-state structures are illustrated by showing in table 12 those for deprotonation of XH;+. For all dications the transition state corresponds to one considerably elongated H-X bond whilst the others remain within ca.0.2 A of their equilibrium values. Deprotonation of the species XHZ+ proceeds through a quasi-planar C S. A. POPE I. H. HILLIER AND M. F. GUEST Table 11. Calculated activation barriers and deprotonation energies (kcal mol-l) deprotonation energy activation barrier HF/TZP CI/TZP CI+ HF/TZP CI/TZP CI+ molecule //TZP //TZP ZPVE //TZP //TZP ZPVE 120.8 117.9 118.4 9.6 8.7 8.1 66.8 57.7 62.0 44.0 42.3 38.6 101.1 105.5 108.6 14.5 13.4 11.4 -202.9 208.6 -0.0 0.0 62.2 66.0 69.0 23.5 19.8 17.0 27.3 20.5 25.9 57.5 65.0 60.9 42.3 48.1 51.6 47.6 42.8 23.1 14.2 16.9 -67.2 64.5 82.7 81.2 84.1 24.7 25.3 23.9 131.6 130.8 131.8 1.6 2.1 1.4 -260.9 262.6 -0.0 0.0 50.7 51.3 53.5 42.3 42.6 40.5 45.0 51.1 61.8 50.9 49.7 39.3 76.6 67.4 69.7 39.3 38.6 36.3 Table 12.Optimized geometrical parameters for the deprotonation transition structures XH;+ -+ XH++ H+ value parametera HF/TZP CASSCF/TZP NH;+(211,)-+ NH+(211)+H+ r(N-H) 1.088 1.114 r(N-H,) 1.824 1.863 WNH 180.0 180.0 0~;+(3zg)-+ OH+(~Z-)+H+ r(0-H) 1.099 1.108 r(O-H,) 1.459 1.553 WOH 180.0 180.0 PH;+(2A,)-+ PH+(211)+ H+ r(P-H) 1.426 -r(P-H,) 2.890 -(HPH 123.7 -SHZ+('A,) -+ SH+(lA)+ H+ r(S-H) 1.370 -r(S-H,) 2.754 -(HSHl 105.8 -~ a H labels the departing proton. 122 STRUCTURE OF HYDRIDES transition state with deviations of < 6" from planarity. Deprotonation of PH:+ which has a C31) ground state proceeds via a C31)transition state the long P-H bond length having increased to 3.437 A.Deprotonation of NH:+(D,,) is predicted to occur through a C structure in which one N-H bond is stretched by 0.3 A. However this transition state is only 8.7 kcal mo1-l above the D2d minimum (reduced to 8.1 kcal molP1 by zero-point corrections). Thus although we have not explained the role of the D, saddle point which lies 7.1 kcal mol-1 above the DBd minimum we conclude that in view of these small barriers NH:+ can have at best a very short lifetime. CONCLUSIONS The principal conclusions resulting from the calculations described herein are as follows. (a) Comparison with experimental geometries reveals a greater underestimation of bond lengths and greater overestimation of bond angles at the HF/TZP level for the first-row than for the second-row hydrides.The subsequent increase and decrease respectively of these quantities at the CASSCF level leads to good agreement with experiment for the first-row species but not for the second-row species indicating an inadequacy in the TZP basis or in the CASSCF space. (b) All the dications studied except OH2+,NH2+,H202+and NHi+ have relatively large activation barriers to deprotonation in qualitative agreement with the results of charge-s tripping experiments. (c) The calculated adiabatic ionization energies for the monocations of nitrogen and oxygen are except for NH significantly larger than experimental values. Calculations for the neutral species yield as expected ionization energies smaller than the experimental values.The discrepancy for the monocations may arise from the involvement of vibrationally excited states of the monocations in the charge-stripping experiment. We thank the S.E.R.C. for support of this research. 1 M. E. Schwartz in Modern Theoretical Chemisrry ed. H. F. Schaefer 111 (Plenum Press New York 1977) vol. 4 p. 358. 2 I. H. Hillier Pure Appl. Chem. 1979 51 2183. 3 W. von Niessen in Molecular Ions Geometric and Electronic Structures ed. J. Berkowitz (Plenum Press New York 1983) p. 355. 4 P. Pulay in Modern Theoretical Chemistry ed. H. F. Schaefer 111 (Plenum Press New York 1977) vol. 4 p. 153. D J. A. Pople in Molecular Ions Geometric and Electronic Structures ed. J. Berkowitz (Plenum Press New York 1983) p.287. 6 R. G. Cooks T. Ast and J. H. Beynon Znt. J. Mass Spectrom. Ion Phys. 1973 11 490. 7 T. Ast C. J. Porter C. J. Proctor and J. H. Beynon Bull. SOC. Chim. Beograd. 1981 46 135. 8 C. J. Proctor C. J. Porter T. Ast P. D. Boltonand J. H. Beynon Org. MussSpectrom. 1981,16,454. 9 T. Ast C. J. Porter C. J. Proctor and J. H. Beynon Chem. Phys. Lett. 1981,78,439. 10 W. J. Hehre and W. A. Lathan J. Chem. Phys. 1972 56 5255; J. S. Binkley J. A. Pople and W. J. Hehre J. Am. Chem. Soc. 1980 102 939. 11 T. H. Dunning Jr J. Chem. Phys. 1971 55 716. 12 A. D. McLean and G. S. Chandler J. Chem. Phys. 1980 72 5639. 13 H. Cardy D. Liotard A. Dargelos and E. Poquet Chem. Phys. 1983 77 287. 14 R. Ahlrichs and P. R.Taylor J. Chim. Phys. 1981 78 315.15 R. Ahlrichs F. Keil H. Lischka W. Kutzelnigg and V. Staemmler J. Chem. Phys. 1975 63 455. 16 R. Ahlrichs F. Driessier H. Lischka V. Staemmler and W. Kutzelnigg J. Chem. Phys. 1975 62 1235. 17 C. W. Bauschlicher H. F. Schaefer and P. S. Bagus J. Am. Chem. Soc. 1977 99,7106. 18 F. Bobrowicz and W. A. Goddard 111 in Modern Theoreticul Chemistry ed. H. F. Schaefer 111 (Plenum Press New York 1977) vol. 3 p. 79. 19 P. E. M. Siegbahn J. Almlof A. Heiberg and B. 0. Roos J. Chem. Phys. 1981 74 2384. S. A. POPE I. H. HILLIER AND M.F. GUEST 123 2o B. 0.Roos P. R. Taylor and P. E. M. Siegbahn Chem. Phys. 1980,48 157. 21 V. R. Saunders and J. H. van Lenthe Mol. Phys. 1983 48 923. 22 B. 0.Roos P. Linse P. E. M. Siegbahn and M. R. A. Blomberg Chem.Phys. 1982 66 197. 23 J. A. Pople H. B. Schlegel R. Krischnan D. J. Defrees J. S. Binkley M. J. Frisch R. A. Whiteside R. F. Hout and W. J. Hehre Int. J. Quantum Chem. 1981 15 269. 24 A. D. Walsh J. Chem. Soc. 1953 2260. 25 S. Bell J. Chem. Phys. 1978 68,3014. 26 B. 0.Roos Int. J. Quantum Chem. 1980 S14 175. 27 R. Krishnan J. S. Binkley R. Seeger and J. A. Pople J. Chem. Phys. 1980,72 650. 2R R. Ditchfield and K. Seidman Chem. Phys. Lett. 1978 54 57. 2g Y. Yamaguchi and H. F. Schaefer 111 J. Chem. Phys. 1980 73 2310. 30 L. E. Sutton Tables of Interatomic Distances and ConJigurations in Molecules and Ions (Special Publication no. 18 The Chemical Society London 1965). 31 G. Herzberg Electronic Spectra of Polyatomic Molecules (Van Nostrand Princeton 1967).32 H. F. Schaefer in Critical Evaluation of Chemical and Physical Structure Information ed. D. R. Lide and M. A. Paul (National Academy of Sciences Washington 1974). 33 M. F. Guest and W. R. Rodwell Mol. Phys. 1976 32 1075 34 H. Lew Can. J. Phys. 1976 54 2028. 35 K. Dressier and D. A. Ramsay J. Chem. Phys. 1957 27 971. 36 R. N. Dixon G. Duxbury M. Horani and J. Rostas Mol. Phys. 1972 22 977. 37 J. M. Berthou B. Pascat H. Guenebaut and D. A. Ramsay Can. J. Phys. 1972 50 2265. 38 G. Duxbury M. Horani and J. Rostras Proc. R. Soc. London Ser. A 1972 331 109. 3g P. J. Bruna G. Hirsch R. J. Buenker and S. D. Peyerimhoff in Molecular Ions Geometric and Electronic Structures ed. J. Berkowitz (Plenum Press New York 1983) p. 309. 4o M. E. Casida M. M.L. Chen R. D. MacGregor and H. F. Schaefer 111 Isr. J. Chem. 1980,19 127. 41 S. K. Shih S. D. Peyerimhoff and R. J. Buenker Chem. Phys. Lett. 1978 55 206. 4z S. D. Peyerimhoff and R. J. Buenker Chem. Phys. 1979,42 167. 43 W. Meyer and P. Rosmus J. Chem. Phys. 1975 63 2356. 44 P. Rosmus and W. Meyer J. Chem. Phys. 1977 66 13. 45 P. J. Bruna G. Hirsch S. D. Peyerimhoff and R. J. Buenker Mol. Phys. 1981 42 875. 46 R. Colin and A. E. Douglas Can. J. Phys. 1968 46 61; M. F. Guest and D. M. Hirst Mol. Phys. 1977 34,1611. 47 P. Helminger F. C. de Lucia and W. Gordy Phys. Rev. A 1974 9 12. F. Y.Chu and T. Oka J. Chem. Phys. 1974,60,4612. 49 S. D. Peyerimhoff R. J. Buenker and L. C. Allen J. Chem. Phys. 1966 45 734; R. J. Buenker and S. D. Peyerimoff Chem.Rev. 1974 74 127. 50 P. W. Payne and L. C. Allen in Modern Theoretical Chemistry ed. H. F. Schaefer 111(Plenum Press New York 1977) vol. 4 p. 29. 51 D. S. Marynick and D. A. Dixon J. Phys. Chem. 1982,86 914. 52 A. Schmiedekamp S. Skaarup P. Pulay and J. E. Boggs J. Chem. Phys. 1977 66 5769. j3 A. Begum A. R. Lyons and M. C. R. Symons J. Chem. Soc. A 1971 2290. 54 J. P. Maier and D. W. Turner J. Chem. SOC., Faraday Trans. 2 1972 68 71 1. 55 L. J. Aarons M. F. Guest M. B. Hall and I. H. Hillier J. Chem. Soc. Faraday Trans. 2 1973 69 643. 56 D. S. Marynick J. Chem. Phys. 1981 74 5186. 57 B. N. McMaster J. Mrozek and V. H. Smith Jr Chem. Phys. 1982,73 131. 58 W. Meyer J. Chem. Phys. 1973 58 1017. 59 J. W. McIver Jr Ace. Chem. Res. 1974 7 72.6o B. W. Williams and R. F. Porter J. Chem. Phys. 1980 73 5598. 61 G. I. Gellene D. A. Cleary and R. F. Porter J. Chem. Phys. 1982 77 3471. 62 G. R. Branton D. C. Frost F. G. Herring C. A. McDowell and I. A. Stenhouse Chem. Phys. Lett. 1969 3 58 1. 63 S. J. Dunlavey J. M. Dyke N. Jonathan and A. Morris Mol. Phys. 1980 39 1121. 64 S. N. Foner and R. L. Hudson J. Chem. Phys. 1981 74 5017. 65 P. G. Wilkinson Astrophys. J. 1963 138 778. 66 J. W. Rabalais T. P. Debies J. L. Berkosky J-T. Huang and F. 0.Ellison J. Chem. Phys. 1974,61 516. 67 S. Katsumata and D. R. Lloyd Chem. Phys. Lett. 1977 45 519. 6B S. J. Dunlavey J. M. Dyke N. K. Fayad N. Jonathan and A. Morris Mol. Phys. 1979 38,729. 6s S. R. Langhoff and E. R. Davidson Int. J. Quantum Chem. 1974,8 61.70 J. A. Pople R. Seeger and R. Krishnan Int. J. Quantum Chem. 1977 S11 149. 'l P. E. M. Siegbahn Chem. Phys. 1982 66,443. 72 J. A. Pople B. Tidor and P. von R. Schleyer Chem. Phys. Lett 1982 88 533.

ISSN:0301-5696    

DOI:10.1039/FS9841900109

出版商:RSC    

年代:1984

数据来源: RSC